MASTER`S THESIS

Transcription

MASTER`S THESIS

CZECH TECHNICAL UNIVERSITY IN PRAGUE
FACULTY OF CIVIL ENGINEERING
MASTER’S THESIS
2012
FILIP DVOŘÁČEK
CZECH TECHNICAL UNIVERSITY IN PRAGUE
Faculty of Civil Engineering
Department of Special Geodesy
Kalibrace elektronických dálkoměrů
Diplomová práce
Calibration of Electronic Distance Meters
Master’s thesis
Study program:
Study field:
Geodesy and Cartography
Geodesy and Cartography
Supervisors:
Ing. Tomáš Jiřikovský, Ph.D.
Prof. Martin Vermeer
(CTU, Prague)
(Aalto University, Helsinki)
Filip Dvořáček
Prague and Helsinki, 2012
CTU,
Aalto, 2012
Abstract and keywords
ABSTRACT
The theoretical part of the paper is concerned with length measuring methods, especially with
electronic distance measurement. Functional principles of electronic distance meters are introduced and velocity corrections to measured distances are described. Further topics deal
with errors and calibrations of instruments, uncertainties of measurement and international
instructions related to calibrations. The practical part of the paper is concentrated on philosophy, realization and interpretation of calibrations of two electronics distance meters at
a calibration baseline called “the Nummela Standard Baseline” in Finland. Last parts compare
calibrations at the Nummela standard baseline and at a similar baseline in Koštice in Czech
Republic.
KEYWORDS
Geodesy, Metrology, Electronic distance measurement, Electronic distance meter, Reflecting
prism, Accuracy, Calibration, Calibration baseline, The Nummela Standard Baseline, Czech
state long distances measuring standard Koštice, Additive constant, Scale correction, Velocity
corrections, Uncertainty of measurement, Leica TCA2003, Kern Mekometer ME5000
ABSTRAKT
Teoretická část práce se zabývá metodami měření délek, zejména pomocí elektronických
dálkoměrům. Uvádí principy elektronického měření vzdálenosti a popisuje fyzikální korekce
měřené délky. Dále se věnuje chybám a kalibracím přístrojů, nejistotám měření a
mezinárodním předpisům týkajících se kalibrací. Praktická část se pak soustředí na metodiku,
provedení a vyhodnocení kalibrace dvou elektronických dálkoměrů na kalibrační základně
Nummela ve Finsku. Závěrečná část pak srovnává proces kalibrace na základně Nummela a
na státním etalonu velkých délek Koštice v České Republice.
KLÍČOVÉ POJMY
Geodézie, Metrologie, Elektronické měření délky, Elektronický dálkoměr, Odrazný hranol,
Přesnost, Kalibrace, Kalibrační základna, Kalibrační základna Nummela, Státní etalon velkých délek Koštice, Součtová konstanta, Násobná konstanta, Fyzikální redukce, Nejistota
měření, Leica TCA2003, Kern Mekometer ME5000
CTU,
Aalto, 2012
Announcement
ANNOUNCEMENT
I declare that I processed this thesis autonomously and that I wrote down all used information
sources in unity with the methodical instruction of CTU in Prague number 1/2009 [1].
This master’s thesis has direct relationship to my previous bachelor’s thesis. Therefore some
parts of this paper are based on [2].
IN PRAGUE, May 2012
……………………………
Filip Dvořáček
CTU,
Aalto, 2012
Thanks and support
THANKS:
For help with the master’s thesis:
Ing. Tomáš Jiřikovský, Ph.D. – supervisor, CTU in Prague
Prof. Martin Vermeer
– supervisor, Aalto University in Helsinki
Lic. Sc. Jorma Jokela
– Nummela administrator, the FGI
Panu Salo
– helping surveyor, Aalto University in Helsinki
Bc. Petra Stolbenková
– helping surveyor, CTU in Prague
Mgr. Veronika Krausová
– text corrections
SUPPORT:
This paper could be written thanks to financial support by the Erasmus fund for students going studying abroad.
CTU,
Aalto, 2012
Contents
CONTENTS
PREFACE................................................................................................................................. 10
1.
2.
INTRODUCTION TO DISTANCE MEASUREMENT ................................................. 12
1.1
Metrology .........................................................................................................12
1.2
Meter as a length unit .......................................................................................13
1.3
Units of temperature, pressure and humidity ...................................................14
1.4
Mathematical reductions ..................................................................................15
1.5
Direct and indirect length measurement ..........................................................16
ELECTRONIC DISTANCE MEASUREMENT ............................................................ 18
2.1
History..............................................................................................................18
2.2
Propagation of electromagnetic waves ............................................................19
2.2.1
The velocity of light ..................................................................................19
2.2.2
The electromagnetic spectrum ..................................................................21
2.2.3
Propagation of different frequencies .........................................................22
2.3
Accuracy ..........................................................................................................23
2.4
Classification of EDM instruments ..................................................................24
2.4.1
Classification according to range ..............................................................24
2.4.3
Classification according to accuracy.........................................................25
2.4.4
Classification according to wavelength ....................................................25
2.5
Principles of the EDM .....................................................................................26
2.5.1
The pulse method ......................................................................................26
2.5.2
The phase difference method ....................................................................27
2.6
The refractive index of air................................................................................28
2.6.1
Barrell and Sears equations .......................................................................29
2.6.2
Error analysis ............................................................................................30
2.6.3
The partial water vapor pressure ...............................................................31
2.6.4
Ciddor and Hill equations .........................................................................32
2.6.5
The coefficient of refraction......................................................................33
2.7
Determination of the refractive index of air ....................................................36
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2.7.1
Normal procedures ....................................................................................36
2.7.2
Limitations of normal procedures .............................................................36
2.7.3
Special procedures ....................................................................................37
2.8
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Velocity corrections to measured distances .....................................................39
2.8.1
The reference refractive index ..................................................................39
2.8.2
The first velocity correction ......................................................................40
2.8.3
The second velocity correction .................................................................41
CALIBRATION OF EDM INSTRUMENTS ................................................................. 42
3.1
Introduction to calibrations ..............................................................................42
3.2
Errors of instruments........................................................................................43
3.2.1
Overview of errors ....................................................................................43
3.2.2
The zero error and the additive constant ...................................................44
3.2.3
The scale error and the scale correction ....................................................45
3.2.4
Other errors ...............................................................................................45
3.3
EDM calibration baselines ...............................................................................46
3.4
Analysis of measurements ...............................................................................48
3.4.1
Simple linear regression ............................................................................48
3.4.2
Separate computation ................................................................................50
3.4.3
Comparison of methods ............................................................................51
3.5
Uncertainty of measurement ............................................................................52
3.5.1
Introduction ...............................................................................................52
3.5.2
Type A evaluation of standard uncertainty ...............................................54
3.5.3
Type B evaluation of standard uncertainty ...............................................55
3.5.4
Standard uncertainty of the output estimate ..............................................55
3.5.6
Expanded uncertainty of measurement .....................................................57
3.6
Instructions related to EDM calibrations .........................................................57
3.6.1
Standards ISO 17123.................................................................................58
3.6.2
Guide to the expression of uncertainty in measurement ...........................59
3.6.3
Document EA 4/02 ....................................................................................60
3.6.5
Document M3003 ......................................................................................61
3.6.6
Standard ISO/IEC 17025...........................................................................61
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3.6.8
Standard ISO 10012 ..................................................................................62
3.6.9
Standards ISO 9000...................................................................................62
CALIBRATION AT THE NUMMELA BASELINE ..................................................... 64
4.1
The Nummela Standard Baseline.....................................................................64
4.1.1
The Finnish Geodetic Institute ..................................................................64
4.1.2
Baseline description ..................................................................................65
4.1.3
Quartz gauge system .................................................................................67
4.1.4
Interference measurements........................................................................69
4.1.5
Projection measurements ..........................................................................72
4.1.6
Scale transfers to other baselines ..............................................................74
4.2
Instruments and equipment ..............................................................................74
4.2.1
The Leica TCA2003 ..................................................................................74
4.2.2
The Kern Mekometer ME5000 .................................................................76
4.2.3
Other used instruments and equipment .....................................................77
4.3
Measurement ....................................................................................................81
4.3.1
Measurement information .........................................................................81
4.3.2
Measurement procedure ............................................................................82
4.4
Evaluation process ...........................................................................................85
4.4.1
Software ....................................................................................................85
4.4.2
Data pre-processing in Excel ....................................................................86
4.4.4
Data processing in Matlab .........................................................................88
4.4.5
Uncertainties of measurement ...................................................................90
4.4.6
Combining results of calibrations .............................................................93
4.5
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Results of calibrations ......................................................................................94
4.5.1
The Leica TCA2003 ..................................................................................94
4.5.2
The Kern Mekometer ME5000 .................................................................98
4.5.3
Summary and final results .......................................................................102
4.5.4
Comparison with FGI results ..................................................................104
NUMMELA AND KOŠTICE BASELINES ................................................................. 106
5.1
The Koštice Baseline .....................................................................................106
5.2
Comparison of baselines, measurements and evaluations .............................108
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Contents
Research and development ............................................................................111
CONLUSIONS ....................................................................................................................... 113
BIBLIOGRAPHY .................................................................................................................. 115
LIST OF USED ABBREVIATIONS ..................................................................................... 118
LIST OF FIGURES ................................................................................................................ 119
LIST OF TABELS ................................................................................................................. 120
LIST OF APPENDIXES ........................................................................................................ 121
LIST OF ELECTRONIC ATTACHMENTS ......................................................................... 122
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Preface
PREFACE
The topic of this paper is calibration of electronic distance meters. The problematic of calibrations of EDM instruments is the essential part connecting terrestrial geodesy and length metrology. Since the days of first length measuring instruments, the aim was to calibrate these
instruments on more accurate calibration standards and etalons to determine the accuracy
traceable to a definition of a length measuring unit. This is the only purposeful way to achieve
unification of measuring distances all over the earth. Today, electronic distance meters are the
most accurate devices for measuring long distances under field conditions. Therefore calibration has become a real challenge demanding use of state-of-the-art technology, instruments
and procedures available.
Because EDM instruments are mostly used under field conditions and the measurement is
strongly dependent on ambient atmospheric parameters, field calibration procedures offer a
way of testing in similar unstable operating conditions. In this case, field calibration baselines
are used to compare distances measured with an EDM instrument with more accurate true
distances of a calibration baseline. Instrumental corrections can then be evaluated which is the
main aim of every calibration. As the principle may seem simple to understand, the whole
traceability chain reaching the definition of the meter is extremely scientific and laborious
when sufficient accuracy is intended to be achieved. Only in several places in the world, calibrations of the most accurate EDM instruments can be performed with satisfactory low uncertainties. With new technologies entering this field of science in the near future, understanding
of present best possible procedures is necessary.
The author of this paper performed calibrations of several EDM instruments on the Czech
measuring standard Koštice as a part of his bachelor’s thesis [2]. Considering the new baseline, measuring instruments, measuring process and limited amount of time, there was a space
for more detailed exploration of problematic of calibrations and for reducing uncertainties. In
this paper, calibrations performed at the Nummela Standard Baseline in Finland are described.
Introductory chapters may seem sort of extensive, but in fact there are the minimum and the
essential parts for understanding the calibration process which is then only an application of
previous pieces of knowledge.
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Preface
The Nummela Standard Baseline is considered to be extremely stable and the most accurate field calibration baseline in the world. It is the only baseline which is regularly measured
with the Väisälä interference comparator with about 0.1 mm/km standard uncertainty. The
accuracy is fully traceable to the definition of the meter through the quartz gauge system and
interference and projection measurements. Two high precision electronic distance meters
were calibrated on this baseline by the author. A Leica TCA2003 with accuracy 1 mm + 1
mm/km declared by the manufacturer and a Kern Mekometer ME5000 with accuracy 0.2 mm
+ 0.2 mm/km. Kern Mekometer ME5000 is known as the most accurate medium range electronic distance meter in the world and very often serves as the scale transfer standard to other
geodetic baselines.
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1. Introduction to distance measurement
1. INTRODUCTION TO DISTANCE MEASUREMENT
1.1 Metrology
A fundamental aim of surveying is the determination of relative positions between two points
in space. To fulfill this aim, necessary measurements of important values must be held. In
geodesy, there are measurements of lengths and angles. Other additional physical quantities
have to be usually measured too (time, temperature, air pressure, humidity, etc.). Results of
geodetic measurement should be generally understandable and useable so it is absolutely necessary that measurements have to be proceeded under unified processes, physical quantities
must be defined and accuracies of measurements must be declared.
Metrology is one of branches of science which deals with these questions. It performs research in fields like physical units and constants, measuring instruments and methods, calibration standards and etalons realisations. A main part of Europe (including the Czech Republic
and Finland) is using a quantity system of physical units called Système international d'unités
(SI system).
In this place, the difference between expressions “accuracy” and “precision” should be
explained to exclude any further misunderstandings. The accuracy of a measurement system
is the degree of closeness of measurements of a quantity to that quantity's actual (true) value.
The precision of a measurement system, also called reproducibility or repeatability, is the degree to which repeated measurements under unchanged conditions show the same results. A
measurement system can be accurate but not precise, precise but not accurate, neither, or both.
See (Fig. 1) and (Fig. 2) below.
Fig. 1: Accuracy vs. precision - graph
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High accuracy but low precision
Low accuracy but high precision
Fig. 2: Accuracy vs. precision - targets
Section 1.1 uses bibliography: [2], [3], [4], [5].
The source of used figures: [5].
1.2 Meter as a length unit
History of determining the length scale is very long. Originally it was related with ancient
trade when there were no unified rules and every bigger city has its own length unit which
was derived from a part of human body (feet), a human activity (step), a natural phenomenon
or used measuring instrument. Wide variety in length units led to efforts to derive a length
unit from the Earth proportions. Several measurements were held to define one degree of the
Earth meridian and the Earth radius.
A first step to international unification of length units came from members of Paris Science Academy Jean Charles Borda, Pierre Simon de Laplace and Gaspar Monge in the year
1791. The length unit called meter was accepted as:
The ten-millionth part of the meridian quadrant of the Earth reduced to sea level.
The result was transferred to a platinum rod with a diameter 25 x 4.05 mm as the final distance 1 m at 0 °C. The name of the length unit was derived from the Greek word metron. Using decimal multiplication and division there was created the metric system of length units.
The larger unit is kilometer [km], the smaller unit is millimeter [mm].
Only 20 states of the world accepted this definition at that time but afterwards more of
others have joined. Metrological Institute in Sévres in France provided length etalon of 1 meter in a form of ”prototype” for each country (Fig. 3). So there was another definition of meter:
Meter is the length between two marks which are in the middle of the prototype in perpendicular position in plane of neutral fibers.
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With this method, accuracy of determination of 1 meter was increased from 1 10-5 to 1 10-7 m.
Fig. 3: The prototype meter bar of the USA
In the year 1960 meter was defined by the wave length of krypton atom:
Meter is length equal to 165076373 of wave length of radiation in vacuum which corresponds
to the transition between levels 2p10 and 5d5 of atom krypton 86.
Today meter is defined by the velocity of light. This definition is valid since 1983.
Meter is the length of path travelled by light in a vacuum during 1 / 299 792 458 of a second.
Section 1.2 uses bibliography: [2], [3].
1.3 Units of temperature, pressure and humidity
For some length measuring methods it is required to measure also other physical quantities
than just distances when high accuracy should be achieved. It is nearly necessary for the
method of electronic distance measurement where velocity corrections must be determined.
Kelvin [K] is the SI unit for temperature.
The kelvin is defined as the fraction 1/273.16 of the thermodynamic temperature of the triple
point of water.
The scale of Celsius [°C] is more often used in practice. The magnitude of the scale line is the
same but the relationship between them is:
0 K  273.15 C
0 C  273.15 K
Temperature is measured by the instrument called thermometer.
The fundamental SI derived unit of pressure is the Pascal [Pa].
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(1.1)
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There is a pressure of 1 Pa in space if there is a force per unit area defined as newton per
square meter.
Other used units are bar [bar] and Torr [Torr]:
1 mbar  1 hPa
1 Torr  1.3332 mbar
(1.2)
Pressure is measured by the instrument called barometer.
Humidity is an essential characteristic of air. We differentiate absolute and relative humidity. Relative humidity is important for determination of air refractive index.
Relative humidity of an air-water mixture is defined as the ratio of the partial water vapor
pressure in the mixture to the saturated water vapor pressure at a prescribed temperature.
Relative humidity is non-dimensional quantity and it is given as a percentage [%].
It is measured by the instrument called hygrometer or it can be computed from psychrometer
measurements.
Section 1.3 uses bibliography: [2], [3], [6].
1.4 Mathematical reductions
The Earth is physical solid created and maintained in its shape by the gravity and eccentric
force. Physical surface is non-homogenous and it is often approximated for geodesy purposes
by mathematically defined surfaces such as a sphere and an ellipsoid. These objects are then
substituted by surfaces, such as a cone and a cylinder, which can be transformed to a plain. By
that approximation some differences in lengths, angles or areas appear.
Length reductions from the shape of the Earth are called mathematical reductions. They
provide a unified determining of distances on the Earth's surface. Among the mathematical
reductions of lengths include sea level reduction and cartographic reduction. To reduce a
length longer than about 10 kilometers is much more complicated because there are other reductions which cannot be ignored as for short lengths.
In case of calibrations of electronic distance meters it is not necessary to use these reductions completely. A calibration baseline is situated on one certain place where the reductions
are always the same values. In addition, we do not need to include these distances into maps.
There is only one reduction which can be used when pillars of a baseline are not in a same
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height. It is the reduction to constant height, most often to a height of the first pillar. Following formula (1.3) can be used for this purpose to obtain horizontal distances in chosen level.
This reduction is often called vertical geometrical correction.
s 2   hT  hS 
ds 
s
 hS  hT 
1   1  
R 
R

2
where
(1.3)
ds … vertical geometrical correction [m]
s
… slope distance [m]
hS … instrument station height above the reference point [m]
hT … target height above the reference point [m]
R … radius of the Earth [m]
Section 1.4 uses bibliography: [2], [3], [7], [8], [9].
1.5 Direct and indirect length measurement
The direct length measurement uses such methods which allow measuring the length directly
without need of other intermediate quantities. Very precise measuring rods were used in the
past. They were 3.9 meters long, made by iron and zinc, and they were placed one after another to the racks which were holding them. Other direct measuring instruments are invar
wires, chains (Fig. 4) and measuring tapes. Measuring tapes are still very often used today.
The methods of direct length measurement could be very accurate if they are done with extreme precision and experience of the measurer. Atmospheric conditions must be measured to
determine changes of measuring instrument. A disadvantage of these methods is the fact that
they are very difficult and time-consuming.
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Fig. 4: A measuring metal chain
By using indirect length measurement length is determined with help of other intermediate quantities. A most simple method is using a measuring wheel with known perimeter and
counting number of its revolutions. In geodesy, there are also several examples of length
measuring instruments using indirect measurement. There are mechanical, optical and physical instruments. Not many of them are still used today. But a method of the parallactic angles
measurement can be still used for very accurate measurements of short distances.
Electronic distance measurement could be covered up by both direct and indirect measurement but more often it is detached separately because of its advanced technology and great
importance. Nowadays it is the most often used method of determining distances in geodesy.
Electronic distance measurement (EDM for short) is described in following chapter 2.
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2. Electronic distance measurement
2. ELECTRONIC DISTANCE MEASUREMENT
2.1 History
Historically, the development of electro-optical distance meters evolved from techniques used
for determination of the velocity of light described in subsection 2.2.1. According to Zetsche
(1979), the first electro-optical distance meter was developed by Lebedew, Balakoff and
Wafiadi at the Optical Institute of the USSR in 1936. The Swedish scientist E. Bergstrand
designed the first “Geodimeter” (geodetic distance meter) for determination of the velocity of
light in 1943. The first commercial instrument Geodimeter NASM-2A (Fig. 5) was produced
by the Swedish company AGA and became available in 1950. Long distances could be only
measured at night.
Fig. 5: Geodimeter NASM-2A
An important development was the introduction of the heterodyne technique to electrooptical distance meters by Bjerhammar in 1954, which enabled the execution of more accurate phase measurements at more convenient low frequencies. The first instrument to employ
the heterodyne principle was the Geodimeter Model 6A. Subsequently the principle was employed in distance meters of all makes.
The use of reflected radiowaves for distance measurements was suggested as early as
1889 by N. Tesla. The first radiowave distance meter (interference principle) was made in
1923. The phase measurement based distance meter (Tellurometer) was developed by T. L.
Wadley at National Institute of Telecommunications Research of South Africa in 1954. HeNe
lasers incorporating infrared emitting diodes were introduced in EDM in the late 1960’s (Wild
Distomat DI10, Tellurometer MA 100, Zeiss SM 11). Further development led to smaller the-
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odolite-mounted distance meters (Kern DM 500, AGA Geodimeter 12, Topcon DM-C2) and
semi-electronic tachometers (Zeiss SM 4, Topcon GTS-1, Sokkisha SDM-3).
The first precision EDM instrument, the Mekometer, was built by K.D. Froome and R. H.
Bradsell in 1961 at the National Physical Laboratory in Tedington (U.K.) and became commercially available early in 1973 as the Kern Mekometer ME 3000. As a successor Kern Mekometer ME 5000 was released in 1986.
The first electronic tachometer (total stations) the Zeiss Reg ELTA 14 was produced from
1970 and it enables electronic readout of distances, vertical and horizontal angles. In 1971 the
AGA Geodimeter 700 followed. Smaller and lighter second generation instruments entered
the market in 1977 and 1978 with the Hewlett Packard HP 3820A, the Wild TC 1, the Zeiss
ELTA 2 and ELTA 4.
In the early 1980’s infrared distance meters using the pulse measurement principle appeared on the market. The first instrument for surveying applications, the Geo-Fennel FEN
2000 was released in 1983. The Wild Distomat DI 3000 followed as a second type of pulsed
infra-red distance meters in 1985.
Section 2.1 uses bibliography: [10].
2.2 Propagation of electromagnetic waves
2.2.1 The velocity of light
The accuracy of an EDM instrument depends ultimately on the accuracy of the estimated velocity of light (or more correctly on the velocity of the electromagnetic wave) in vacuum resp.
in the atmosphere. The first estimates of the velocity of light were derived from astronomical
observations because very large distances were necessary to produce effects measureable at
that time. By studying the times of the eclipses of the satellites of Jupiter, Romer first calculated the value to be 187 000 miles per second in the year 1676. Bradley, in 1727, improved
the value by using observations made on the aberration of light received from stars. He obtained the value of 308 000 km/s by measuring the diameter of apparent circle of orbit.
In 1849, A. H. L. Fizeau succeeded in making first direct terrestrial measurement. His
method is shortly described here as a fundamental principle of determining the velocity of
light. The description also demonstrates (Fig. 6). The light source is brought to a focus in the
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plane of the toothed wheel using the lens and the semi-silvered mirror. Rotation of the toothed
wheel causes the light beam to be chopped up into pulses which are then collimated into a
parallel beam by another lens. At the distant station about 8.63 km away, the light is reflected
back to the instrument by means of the mirror and the lens. The pulses of light, when not cut
off by the wheel, could be seen. Now, if the rotation of the wheel is such that the pulse of
light produced by the first gap travels the double light path in such a time that on arrival back
at the wheel the first tooth is in front of aperture, then no light will be seen by the observer.
Fizeau’s 720 teeth cogwheel was spin with angular velocity of 12.6 revolution per second and
313 000 km/s was the estimated velocity of light.
Fig. 6: Fizeau’s experiment
Later, Foucault employed a rotating mirror in 1862 and Michelson a rotating prism in
1927 for similar experiments. Michelson used a mile-long tube reduced to conditions of low
vacuum in order to minimize errors due to atmospheric effects and achieve a value of 299 774
km/s. By 1940, a whole series of experiments had been carried out and Birge computed a
weighted mean value for the velocity of light as 299 776
4 km/s.
During the Second World War, radar devices using microwaves were introduced into
range-finding techniques. Repeated use of these instruments over known distances shows significant difference between visible light and microwaves measurements. Further experiments
using Kerr cell and Geodimeter and Tellurometer instruments were in close agreement to the
value determined by microwave instruments and cavity resonators. In 1973, the Consultative
Committee for Definition of the Meter recommended a value of 299 792 458
use in all practical applications.
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So does the uncertainty of the velocity of light in vacuum effects geodetic measurements?
The standard deviations of the velocity of light amounts only 0.004 ppm which is significantly smaller value then EDM instruments can achieve. Since 1983, meter is defined as the
length of the path travelled by light in a vacuum in 1/299792458 of a second. As a result of
this definition, the velocity of light in vacuum is exactly 299 792 458 m/s and has become a
defined constant in the SI system of units.
Subsection 2.2.1 uses bibliography: [10], [11], [12].
2.2.2 The electromagnetic spectrum
The electromagnetic spectrum (Fig. 7) is the range of all possible frequencies of electromagnetic radiation. There is an elementary equation (2.1) describing relationship between a
frequency of a signal f and a wavelength in a medium

in which light has a velocity v.
v
f
Fig. 7: A diagram of the electromagnetic spectrum
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Electromagnetic radiation can be described by the following equation (2.2).
where
y
y  A sin  t   A sin   
(2.2)
 t
  2 f
(2.3)
… instant strength of the wave
A … amplitude
 … angular velocity (angular frequency)
f
… frequency of the signal
t
… time
 … phase angle
2.2.3 Propagation of different frequencies
The mode and the velocity of propagation of an electromagnetic wave are dependent on
the frequency of the signal and the nature of the Earth’s atmosphere. EDM instruments are
capable of measuring path lengths to a very high degree of accuracy, but this information
cannot be fully utilized unless the nature of that path is understood.
Although EDM instruments use only part of the spectrum from long-waves to visible
light, there are some significant differences between propagation of lower frequencies below
about 30 MHz and higher frequencies above this value. A low-frequency signal has a direct
wave, a ground wave capable of trans-horizon distances, and a reflected wave coming from
the ionosphere. Long-range EDM instruments use the ground wave, the reflected wave can
cause errors in the measurement. Higher frequency signal has a direct wave, a reflected wave
from the ground surface and a scatter signal caused by the atmosphere. EDM instrument using
the higher frequencies all use the direct wave, therefore they are limited in range to something
less than 100 km. The ground reflection is a nuisance and it is possible source of errors in
measurements.
In vacuum, all electromagnetic waves propagate with the same velocity, which is the velocity of light. In the atmosphere, the velocity must be always something less than this value
due to a density of the atmosphere. The velocity of a signal in a medium (the atmosphere) is
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also a function of the frequency of the signal. A single electromagnetic wave propagates with
a phase velocity. In a real EDM instrument, there is a bunch of near frequencies (also modulated) which are formed in a band propagating through the atmosphere. Each frequency travels with a different velocity and the interference between them occurs. The signal resulting
from the sum of all frequencies will have the group velocity which is always smaller than the
phase velocities of its individual frequencies.
Subsection 2.2.3 uses bibliography: [10], [11].
2.3 Accuracy
In this place, it is necessary to introduce how the accuracy of EDM instruments is mostly often presented because it is important for understanding of following sections. An accuracy of
each EDM instrument used for geodetic purposes should be originally declared by its manufacturer. The accuracy of a measured distance is set by the equation (2.4).
md  a  b d
where
(2.4)
md … standard deviation of a measured distance [mm]
a
… constant [mm]
b
… constant [mm]
d
… measured distance [km]
It means that the accuracy of the measured distance contains the fixed part and the variable
part which is dependent on the measured distance. Constants a and b have characters of
standard deviations. Manufacturers should determine these constants according to valid international standards. The constant a is mostly a value between 0.1 and 10 mm, the constant b is
mostly a value between 0.2 and 20 mm.
More often, following expression (2.5) is used for describing the accuracy of an EDM instrument in general.
m  a mm  b ppm
(2.5)
Ppm stands for Latin expression pars per milion, resp. English expression parts per million.
For example the value 2 mm + 2 ppm means that the accuracy of the EDM instrument is 2
mm plus 2 mm for each kilometer of a measured distance. The accuracy of the distance of 2
km measured with such EDM instrument would be 6 mm.
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Consider that the constant a includes resolution of the instrument, short periodic errors,
non-linear distance-dependent errors, accuracy of initial additive constant and compatibility of
reflectors. Constant b includes the drift of the main oscillator within the specified temperature
range and the resolution of ppm settings (see section 3.2). This is a mix of random and quasirandom errors with systematic errors which are impossible to separate.
The accuracy of an EDM instrument can differ significantly according to specific type
and model of the instrument. That is why it is necessarily important for each measurer to
know the accuracy of the instrument. Values of constants determined by the manufacturers
are not always reliable enough and they are not durable in time. The instruments can achieve
significantly better results than which the manufactures declare when they are proper calibrated. Parameters of the EDM instruments can change by reason of improper using, transportation, storage, and age of the instrument. Calibration process serves for checking and rectifying
instruments.
2.4 Classification of EDM instruments
2.4.1 Classification according to range
Three classes of instruments may be distinguished with respect to range:
1. short range EDM instruments
(up to 2 km),
2. medium range EDM instruments (up to 10 km),
3. long range EDM instruments
(up to 100 km).
Short and medium range instruments are usually used for survey praxis. Long range instruments may have a larger minimum distance. Distances up to 70 km may be measured on clear
days with great visibility only, distances up to 100 km and possibly above 100 km may be
measured at clear nights. The range does not depend only on an EDM instrument but also on a
reflector. Mini-prism, 360° prism, standard prism and several or many joined prism integration, they each reflect different amount of signal and thus affect the final range.
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2.4.3 Classification according to accuracy
Four categories of instruments may be distinguished with respect to their accuracy. The
accuracy is described by the standard deviation of one observation:
1. greater than (5 mm + 5 ppm),
2. equal or smaller than (5 mm + 5 ppm) and greater than (1 mm + 2 ppm),
3. equal or smaller than (1 mm + 2 ppm), and greater than (0.2 mm + 0.2 ppm),
4. equal or smaller than (0.2 mm + 0.2 ppm).
Group 1 includes a number of old pulse distance meters and distance measurements to passive
or moving targets. The second group of instruments includes most of distance meters found in
general surveying practice. Group 3 instruments are often referred to as precision distance
meters. Group 4 can be called high precision distance meters and includes the instrument
Kern Mekometer ME5000.
2.4.4 Classification according to wavelength
EDM instruments may use various frequencies resp. wavelengths of the carrier signal, but not
the whole electromagnetic spectrum is suitable for accurate geodetic use. Instruments can be
divided into following groups:
1. instruments using long radio waves,
2. instruments using microwaves,
3. instruments using visible light,
4. instruments using infra-red radiation.
Long radio waves instruments use the ground wave and suffer from interference from an
unwanted sky wave. All can operate over long ranges (hundreds of kilometers) and transhorizon distances, but are not capable of high accuracies (only several meters) because of the
surface phase lag effect. The effect can predicted over a homogenous surface such as water,
that is why these instruments are used for marine navigation purposes and oceanographic surveys.
Group 2 instruments use wavelength from about several centimeters to about several millimeters. Disturbing ground reflection called ground swing is experienced with these instruments because of high divergence of the beam. The range is about 150 km and the accuracy
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which can be achieved is about 1 cm. These instruments were used in geodesy in the past but
now there are not used anymore.
Instruments belonging to group 3 and 4 are instruments used in general geodesy nowadays. The direct wave is used for measuring and unwanted reflections are very uncommon as
most surfaces found in nature do not produce a strong reflection at this wavelength. The beam
is very well collimated. The range is limited by direct visibility, the power of the source and
atmospheric conditions. The accuracy of these instruments may be less than 1 mm but the
accuracy for longer distances is limited by determining atmospheric conditions.
2.5 Principles of the EDM
There are several principles of electronic distance measurement but only some of them
are related to terrestrial geodetic measurement. The pulse method and the phase difference
method are the ones. The Doppler methods and interferometry are used only for special purposes. The meaning of this section is not to describe the functionality in details, but just provide quick overview of ideas which enable to perform electronic distance measurement.
2.5.1 The pulse method
The pulse method is not easy to realize with sufficient accuracy but it is very simple method
to understand the principle. A short, intensive signal is transmitted by an instrument. It travels
to a target point and back and thus covers twice the distance. By measuring the so-called timeof-flight between transmission and reception of the same pulse, the distance may be calculated as (2.6).
d
where
c
v
 t´  t´
2n
2
d
… distance [m]
c
… velocity of light in vacuum [m/s]
n
… refractive index of ambient air
t´ … time-of-flight [s]
v
… velocity of light in ambient air [m/s]
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(2.6)
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The flight time is measured by the oscillator. Considering the velocity of light, an extreme accuracy of measuring the flight time is required. To obtain the precision of 3 mm in a
measured distance, the precision of 20 picoseconds in time-of-flight is required. The pulse
method is frequently used in passive reflecting systems when the pulse is reflected directly
from the surface and no reflecting prism is necessary. Some disturbing reflections can be
identified and filtered out while analyzing the received pulses.
2.5.2 The phase difference method
Historically most important principle which enabled EDM instruments to measure with
high accuracy is the phase difference method. Still a lot of today’s instruments use it, regardless of whether they use light waves, infrared waves or microwaves as carrier waves. The
measuring signal, which is modulated on the carrier wave in the emitter, travels to the reflector and back to the EDM instrument, where it is picked up by the receiver.
In the receiver, the phase of the emitted and the received signals are compared and the
phase lead is measured. The emitted signal can be expressed as (2.7), the signal with a phase
lead (  ) and corresponding time lead ( t ) can be expressed as equation (2.8).
yE  A sin  t   A sin 
(2.7)
yR  A sin       A sin   t  t 
(2.8)
Because a continuous signal is used, the values of y change with time, but the phase lead and
time lead remain constant. Distance cannot be computed as simple as (2.6) because information about flight time is not obtained through phase comparison.
Another equation (2.9) is employed to solve this problem.
t´ mt1  t
where
t´ … flight time of a specific signal [s]
m … number of full wavelengths over the path (ambiguity)
t1
… elapsed time for one full cycle of the modulation signal (period) [s]
t … time lead in phase measurement [s]
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(2.9)
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Although the ambiguity is still unknown, other variables can be computed from (2.10).
t 
 

; t1 
2 v
v
(2.10)
The equation (2.6) can now be written in a new form (2.11) considering relations above.
d
v
v     
  
 mt1  t    m 
m 
2
2  v 2 v 
2 2 2
(2.11)
Usually, the term (2.11) is simplified by using substitutions as follows (2.12).
d  mU  L
where
(2.12)
U … unit length of a distance meter [m]
L … fraction of the unit length U (determined by phase measurement) [m]
The ambiguity is solved by the introduction of more than one unit length in an EDM instrument. The precision of an instrument depends on the choice of the main unit length, because of the limited resolution of the phase measurement.
2.6 The refractive index of air
The phase refractive index ( n ) of a medium is defined as (2.13).
n
c
vm
(2.13)
So it is a ratio of the velocity of light in vacuum ( c ) and the phase velocity of the electromagnetic wave in a medium (
). The variation of the refractive index with frequency causes
the phenomenon of dispersion.
In EDM equipment we are concerned almost exclusively with the refractive index of air. The
refractive index of air is a function of:

the gaseous composition of the atmosphere (which is very nearly a constant),

the amount of water vapor in the atmosphere,

the temperature of the gaseous mixture,

the pressure of the gaseous mixture,

the frequency of the radiated signal.
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The refractive index in a single point in space can be determined by the process described
in this section below. The most important is to realize that the refractive index is changing
with a position on the earth’s surface. So it is also changing along the path of an electromagnetic wave resp. along a measured distance. In the section 2.7 it is described how to deal with
this problem.
2.6.1 Barrell and Sears equations
The group refractive index ( ng ) is determined rather experimentally than mathematically.
Following formula (2.14) according to Barrell and Sears (1939) was derived from experiments at wavelengths (  ) between 436 and 644 nm and it was adopted by IUGG in Berkley
in 1963.
n
g
 1.6288 
 0.0136 
 1 106  287.604  3  
  5

2
4
  
 

(2.14)
It is valid for visible light in standard conditions (dry air, temperature 0 °C, pressure 1013.25
hPa, 0.03 % CO2).  is the “effective” wavelength in vacuum in micrometers. Equation
(2.14) describes the group refractive index with an accuracy of
0.1 ppm or better. The for-
mula can be used to calculate the group refractive index for wavelength to 900 nm (infra-red
radiation) with the similar accuracy. The error of 5 nm in the wavelength causes an error of
0.3 ppm in the group refractive index.
As we know the group refractive index for standard conditions, the group refractive index
for ambient conditions has to be computed. Original formula (2.15) given by Barrell and
Sears and simplified version (2.16) are displayed below.
 nL  1   ng  1
273.15 p
11.27 106

e
 273.15  t  1013.25 273.15  t
ng  1
p
4.125 108
nL  1 


e
1   t 1013.25
1  t
where
nL … ambient group refractive index for atmospheric conditions
t
… temperature of air [°C]
p … atmospheric pressure [hPa, mbar]
 … coefficient of expansion of air (  1/ 273.15 per °C)
e
… partial water vapor pressure [hPa, mbar]
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(2.15)
(2.16)
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According to this formula, the group refractive index for standard constructional conditions
(reference conditions) and for ambient conditions (conditions of measurement) can be computed. Edge (1960) stated a maximum error of the above equation (2.15) to be 0.2 ppm for
temperature range ( 40;
50) °C and pressure range (533; 1067) Torr. Some systematic er-
rors of these equations were discussed and more complicated processes were developed to
eliminate small systematic errors when they are not tolerable in the most precise electronic
distance measurement.
2.6.2 Error analysis
Error analysis of the formula above (2.15) is very important for understanding the importance of measuring atmospheric conditions. The errors of t , p, e may be analyzed by their
partial differentials. For a temperature of
15 °C, a pressure of 1007 hPa, a partial water va-
por pressure of 13 hPa and a group refractive index of 1.0003045 (a visible light EDM) differentiation of (2.15) looks like (2.17).
dnL 106  1.00 dt  0.28 dp  0.04 de
(2.17)
Notice that an error of a refractive index affects a measured distance by the same value in
ppm. The result can be summarized as follows:

An error in t of 1.0 °C affects a distance by 1.0 ppm.

An error in p of 1.0 hPa affects a distance by 0.3 ppm.

An error in e of 1.0 hPa affects a distance by 0.04 ppm.
In other words, to obtain an accuracy of 1 ppm of a measured distance, it is required to measure the three quantities with following accuracies (the other two quantities are considered errorless each time):

t with accuracy of 1 °C,

p with accuracy of 3 hPa,

e with accuracy of 25 hPa.
Very often, the effect of the partial water vapor pressure is omitted in formulas provided
by EDM manufactures. Ignoring the last minuend of the expression (2.15) causes the following errors (Table 1) of a refractive index resp. a measured distance.
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Table 1: Distance errors caused by omission of humidity
Temperature
[°C]
0
10
20
30
40
50
where
h
Errors [mm/km]
h  50 % h  100 %
0.1
0.2
0.2
0.5
0.4
0.9
0.8
1.6
1.4
2.8
2.3
4.6
… relative humidity [%]
2.6.3 The partial water vapor pressure
The partial water vapor pressure can be computed from psychrometer measurements.
Psychrometer consists of two thermometers, one of which is dry and one of which is kept wet
with distilled water. Evaporation of water circulates around the bulb with help of the ventilator and lowers the temperature of “wet bulb” thermometer. The amount of evaporation depends on the humidity of air. From readings of these “wet bulb” and “dry bulb” temperatures,
the partial water vapor pressure can be derived (2.18). This formula is accurate to
1 % for
°C.
e  E´ A p  t  t´
where
e
(2.18)
… partial water vapor pressure [hPa, mbar]
E´ … saturation water vapor pressure for t´ [hPa, mbar]
A … psychrometer constant [°C-1] (Assman psychrometer: A  0.000662 C1 )
p … atmospheric pressure [hPa, mbar]
t
… dry bulb temperature [°C]
t´ … wet bulb temperature [°C]
The values for E´ can be taken from some pre-filled table or can be computed according to
Buck (1981) from equation (2.19). Maximum relative error of E´, computed according to
(2.19), is less than 0.20 % for
20 °C and
50 °C.
17.502 t ´
E´ 1.0007   3.46 106 p  6.1121 e 240.97t´
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(2.19)
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E´ … saturation water vapor pressure for t´ [hPa, mbar]
t´
… wet bulb temperature [°C]
p
… atmospheric pressure [hPa, mbar]
Another way to determine the partial water vapor pressure is to compute it from relative
humidity (2.20).
e
where
Eh
100
(2.20)
E … saturation water vapor pressure [hPa, mbar]
h
… relative humidity [%]
E is calculated similar to E´ (2.19) but using dry bulb temperature ( t ) instead of wet bulb
temperature ( t´ ). Sufficient accuracy about
0.05 ppm can be achieved for nearly all visible
light and infra-red frequency measurements if the relative humidity is measured with the accuracy of
3 % and there is the temperature about
10 °C, there is even lower error about
30 °C. When temperature is about
0.02 ppm. In case of microwave EDM instruments,
the use of measured relative humidity causes a significant loss of accuracy in most cases.
All calculations above assume that water vapor do not condensate in the air. If it is raining, there are no parts in refractive index formulas which can take it into account. According
to experiments, light rain causes a minor error of several hundreds of ppm of a computed refractive index of air. But heavy rain causes a significant error of several tenth of ppm.
2.6.4 Ciddor and Hill equations
Although the Barrell and Sears equations are the best known and still very often used equations, there are newer and more accurate formulas for determining the refractive index of air,
for example equations by Edlén, Owens, Peck and Reeder, Jones. Recent measurements
proofed significant errors of Barrell and Sears formulas, for example underestimated influence of humidity. At 32nd IAG General Assembly in Birmingham in 1999, a resolution about
new formulas according to Ciddor and Hill [14] was adopted because of following reasons:
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
improved accuracy of EDM instruments,

new absolute and relative measurements of the refractive index of air,

the international temperature scale was revised in 1990,

a carbon dioxide content of air has changed since 1963.
Following simplified formulas (2.21) and (2.22) are recommended for use in computations of group refractive index of visible light and near infrared EDM instruments. These
closed formulas deviate less than 0.25 ppm from the accurate formulae between
30 °C and
45 °C, at 1013.25 hPa and 100 % humidity.
N g   ng  1 106  287.6155 
4.88660

2

0.06800
4
 273.15 N g p  11.27 e
N L   nL  1 106  


T
 1013.25 T 
where
(2.21)
(2.22)
N g … group refractivity for standard air
N L … group refractivity for ambient air
T
… thermo-dynamical temperature [K], T  273.15  t
Other variables have the same meaning as noted in section 2.6.1. Standard air parameters are
the same as for Barrell and Sears equations with the exception of CO2 concentration which is
now 0.0375 %.
2.6.5 The coefficient of refraction
It is known from the physics theory that the refractive index of air not only affects the velocity
of light but also the geometry of its path. When a wave passes through regions of differing
refractive index, the wave path will differ slightly from a straight line. It is because an electromagnetic wave always propagates through a path which takes a minimum time to pass
(Fermat’s principle). And it is such a path where the refractive index in each region is the
lowest of all regions around. The effect is different for each wavelength because of its different group refractive index.
The coefficient of refraction is defined as a ratio of a curvature of a wave path and a curvature of a spheroid along a line (the curvature of the earth’s surface).
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k
where
1/ r R

1/ R r
k
… coefficient of refraction
r
… radius of the curvature of the wave path [m, km]
(2.23)
R … radius of the curvature of the spheroid (radius of the Earth) [m, km]
The coefficient of refraction is non-dimensional value. The curvature of the wave path itself
can be defined as (2.24).
1
1  n 
1  n 
  sin z      
r
n  h 
n  h 
where
(2.24)
n
… vertical gradient of the refractive index of air
h
z
… zenith angle between the direction of the gradient of the refractive index
and the tangent to the wave path
The change of the refractive index with height is caused by the vertical density gradient of air.
Gauss experimentally enumerated the coefficient of refraction for visible light waves and
for long-length lines high above ground to the value of 0.13 and this value is very often used
for calculations with refraction. But this value may vary considerably under differing atmospheric conditions, for example during night, at sunrise, at sunset and even during the whole
day. The variation of the coefficient of refraction increases when the line of sight becomes
more close to the ground. For short range EDM instruments the value of 0.13 is worthless for
most cases. Following figure (Fig. 8) illustrates the range of values of the coefficient of refraction during one day (Central Europe, summer, clear, calm wind) for different heights of
lines above ground (according to Hübner, 1978).
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Fig. 8: A model of the daily cycle of the coefficient of refraction
The coefficient of refraction can be determined by measuring the vertical gradient of refractive index resp. at least vertical temperature gradient. One method of eliminating the influence of the refraction on the measurement is to perform the measurement when there is a
minimum of detected refraction resp. when there is a minimal vertical temperature gradient.
The value can be also obtained by simultaneous reciprocal zenith angle measurement. But all
the techniques of determining the coefficient of refraction require a lot of technical equipment
and considerable amount of time. The difference in path length between the theoretical path
(line) and more real path (arc, curve) is very small and can be ignored in most cases. This
statement is not true for microwave EDM instruments and long distances. The refraction is
much bigger problem while measuring zenith angles.
Section 2.6.5 uses bibliography: [10], [11].
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2.7 Determination of the refractive index of air
In principle, electronic distance measurements should be corrected for the integral refractive
index over the entire wave path. This could involve the measurement of temperature, pressure
and humidity along the wave path to calculate it. In practice, simplified methods of measurement are usually adopted. However these methods have their limitations. There are also some
special methods how to eliminate the refractive index.
2.7.1 Normal procedures
The normal procedure of determining the refractive index of air is based on calculations
of the representative refractive index of a line computed from meteorological observations at
one or both terminals of the line. Whenever high accuracy is required, atmospheric conditions
should be measured at both endpoints of the line and the refractive indexes should be computed separately for both terminals. The mean refractive index is then used to correct the measured distance. This procedure provides the best results when only the metrological
measurements at the endpoints are considered.
There is a way to simplify both the calculation and the measurement. Calculations are
simplified by taking the mean of the endpoint meteorological measurements before calculating the representative index of the line. Small errors are introduced in the process due to the
non-linearity of the refractive index equations. Only when there are high differences between
atmospheric conditions at both endpoints, the error could gain considerable value. Differences
of about 7 °C and 100 hPa cause the error of 0.2 ppm.
The measurement could be simplified by measuring the metrological data only at the instrument station. Sometimes also humidity is not taken into account. Only these meteorological measurements are then used to calculate the refractive index at the instrument station. As a
result of this, temperature and pressure variations over the line are ignored. For example because of the change of pressure with height, systematic error approximately 1.6 ppm per each
100 m height difference between the instrument and the reflector stations is introduced.
2.7.2 Limitations of normal procedures
Refractive indexes evaluated from temperature, pressure and humidity readings taken at endpoints represent the integral value over the wave path reasonably well as long as the wave
path is parallel to the ground. If the average height of the line above ground exceeds the in-
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strument and the reflector heights significantly, the mean refractive index no longer closely
approaches the integral value over the wave path. The temperature and humidity measurements at the endpoints are strongly influenced by the closeness of the ground. This is called
ground proximity effect.
The ground proximity effect is the worst during clear sunny days and clear nights. During
sunny days there is an unstable stratification of air when there is an upward heat flux. Reduced ground proximity effect is experienced during neutral stratification with no heat transfer between ground and air (zero heat flux) and zero temperature gradient. Near-neutral
conditions can be typically observed 1.5 hour after sunrise and 1.5 hour before sunset. Neutral
conditions also prevail in heavily overcast weather with moderate to strong wind. Generally
speaking, the cloud cover over the wave path and wind significantly reduces the ground proximity effect.
All discussed weather conditions above were conditions of homogenous weather along
the wave path. Even when there is a linear drift of atmospheric conditions between endpoints
of the line, the mean refractive index describes this change very well. However in some cases
there are radically different conditions in some regions of the wave path. For example when
the wave propagates through areas of direct sunlight and heavy shadows (e.g. wood and
meadow) or when it propagates above surfaces with different temperatures (e.g. land and water), the computed mean refractive index is quite an unreliable value.
2.7.3 Special procedures
Special procedures were developed to overcome limitations of normal procedures. Special procedures involve instrumental solutions, modeling of the atmosphere (rectification
models) and modified field and computing procedures (operational models). Because these
methods are not widely used, only short description will be given here.
A simple solution for discovering, but much more complicated for realization, is to perform atmospheric measurements along the wave path. It means atmospheric data have to be
collected in discrete points of suitable distances between each other along the path. When the
line goes near to the ground, only additional measuring instruments and measurers are demanded. When the line goes far above the ground, suitably equipped planes, motor gliders or
helicopters are required to carry the airborne instruments. That is why the method is extremely expensive.
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Another method, which seems to be very effective and less demanding, is to use multiple
wavelength EDM instrument. Simultaneous distance measurements with two or three different carrier waves take advantage of the dispersion effect and permit the determination of the
difference between two differential refractive indexes. This method leads to a greatly reduced
dependency on accurate determinations of atmospheric parameters. In the case of “twocolour” instrument, the distance is derived as following equation (2.25).
D  dred  A  dblue  dred 
where
(2.25)
D … corrected distance [m]
d red … distance measured with red laser (HeNe) [m]
dblue … distance measured with blue laser (HeCd) [m]
A … coefficient dependent on wavelength, pressure and water vapor pressure
Due to the coefficient A  21 , the distance difference has to be measured 20 times higher resolution than required in the final distance. But surprisingly good result can be achieved. Experiments showed accuracy better than 0.12 ppm. Unfortunately none multi-color EDM
instrument is nowadays commercially available.
The endpoint measurements of temperature, pressure and humidity are representative for
the atmospheric layer close to the ground. To make these measurements representative for the
wave path high above the ground, they need to be transformed. Large number of atmospheric
rectification models was developed to describe the low atmosphere. To apply these models,
additional information such as height profile, wind speed, sun elevation angle, heat flux,
evaporation etc. have to be known. Most models are determined to be used under certain conditions. A suitable atmospheric rectification model certainly determines the mean refraction
index with better reliability than no atmospheric model, but the final accuracy is a big question in most cases.
There is a direct way how to determine the ambient refractive index directly without
measuring the atmospheric conditions. We can obtain refractive index by using instrument
called refractometer, which is mostly dual-laser interferometer. The first beam is travelling in
vacuum, the second beam in ambient air and the change of the optical length can be measured. Refractometers cannot overcome the limitations of atmospheric measurements and they
are not a huge contribution to their accuracy either. However, they are ideally suited for indoor precision EDM instruments and interferometry where there is a uniform but non-
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standard composition of air. Standard refractive index formulas do not reflect special composition of air.
The length ratio method as an operational model solves the problem in a completely different way. When only relative changes are monitored in some geodetic networks and absolute scale is not critical, the measured lines are evaluated by ratio numbers according to their
distances. Groups of distances are assigned a common unknown scale parameter. In the adjustment, scale is evaluated only by defining fixed coordinates of two points in network. This
method really does not deal with the absolute scale problem but sometimes it can be the
cheapest and most efficient solution.
2.8 Velocity corrections to measured distances
When it is clear how to compute and how to measure the refractive index of air, it is necessary
to know how to apply the correction to a measured distance. Modern total stations can do it
instead of us by computing it in an internal firmware. But a user should know how it is done
and someone could go through this process manually because of various reasons. For example
when humidity cannot be the input to the instrument’s software and you want to take it into
account, it is necessary to adopt these computations. Higher accuracy and reliable comparison
of results between several instruments can be achieved.
2.8.1 The reference refractive index
The reference refractive index is the instrument-specific. It is the specific refractive index for
which the EDM instrument provides a correct readout of a distance and it is defined by (2.26).
nref 
where
c0
mod  f mod
nref … reference refractive index
c0 … velocity of light in vacuum [m/s]
mod … constant modulation wavelength [m]
f mod … constant modulation frequency [Hz]
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The reference refractive index is fixed by the manufacturer by adjusting the main oscillator to
such a modulation frequency that then the fixed nref corresponds to an average refractive index encountered under field conditions. When field conditions are exactly same as reference
conditions than n  nref and no refractive index corrections have to be computed.
If such details as the modulation wavelength and the modulation frequency are not available for the specific instrument, the reference temperature, pressure and humidity have to be
available to compute reference refractive index. Reference conditions are such conditions
which correspond to the reference refractive index. It can be computed according to (2.21)
and (2.22).
2.8.2 The first velocity correction
If field measurements are not performed under conditions which are the same as reference
conditions, the first velocity correction must be considered. Different distances can be computed as follows (2.27).
d´
where
c0 t´
c t´
; d 0
nref 2
n 2
(2.27)
d ´ … distance corresponding to reference conditions [m]
d
… distance corresponding to ambient conditions [m]
t´ … measured “flight” time of the signal [s]
n
… refractive index for ambient conditions
And so the rigorous relationship between distances d and d ´ is the equation (2.28).
n
d   ref
 n

 d´

(2.28)
However, it is usually preferred to work with a differential equation. The first velocity correction ( K´ ) is then derived according to following process (2.29).
K´ d  d´ c0
t´  1 1  c0 t´  nref  n 
 nref  n 
 


  d´

2  n nref  2 nref  n 
 n 
(2.29)
Sometimes the first velocity correction is formed as (2.30) when the denominator is set to be
equal to 1. This simplification does not introduce error in excess of 0.02 ppm.
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K´ d´ nref  n 
(2.30)
And then the corrected distance is computed as (2.31).
d  d´ K´
(2.31)
2.8.3 The second velocity correction
As mentioned earlier, atmospheric observations are taken normally only at instrument and
reflector stations and refractive indexes are computed for both terminals of the line. The mean
refractive index is then used for correcting the distance. It would be correct if the wave path
would have the same radius as the radius of the earth’s surface. If there is a geodetic refraction expressed with a coefficient of refraction (section 2.6.5) and second velocity correction is
not used, a small error is introduced into the distance.
Assuming linear vertical gradients of temperature and pressure, the mean refractive index
must be corrected as (2.32).
n1  n2
n1  n2
d´2
2
n
 n 
 k  k 
2
2
12 R 2
where
n
(2.32)
… corrected value of mean refractive index
n … correction of the mean refractive index
k
… coefficient of refraction
R … mean radius of curvature of the spheroid along the line [m]
The second velocity correction ( K´´ ) may be written as (2.33).
K´´ d´ n    k  k 2 
d´3
12 R 2
(2.33)
The second velocity correction is more important for microwaves than for light waves and
more important on long distances than on short distances. The standard coefficient of refraction 0.13 is very often enumeration of the equation but it is unreliable value in most cases.
Computed coefficient of refraction should be used if possible.
The final corrected distance is obtained with equation (2.34).
d1  d´ K´ K´´
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3. Calibration of EDM instruments
3. CALIBRATION OF EDM INSTRUMENTS
This chapter should introduced reasons for calibration and some parts of the calibration process itself. Information here is taken from referred bibliography as well as it is inspired by
author’s experiences with calibrations of electronic distance meters.
3.1 Introduction to calibrations
The calibration of a distance meter is defined as the determination of its instrumental correction and associated accuracy. The instrumental correction is added to a measured distance to
obtain the true value of the distance. The instrumental correction is a function of a number of
independent variables where the most important is the distance itself, atmospheric conditions
and time. To determine instrumental correction, the set of instrument and reflector must be
calibrated together. When using another type of reflector, a new calibration is required.
Electronic distance meters may be calibrated for a number of reasons. The most important ones are:

quality control (at time of purchase and periodically afterwards),

improvement of accuracy (achieving more reliable results),

legal metrology (required by state or international law).
The quality control measurements at the time of purchase have to establish if an instrument
fulfills the manufacturer’s specifications such as accuracy, temperature range and distance
range. The specified temperature range for the operation of the instrument cannot be easily
verified by users. When accurate measurement near the edge of the temperature range must be
performed, the user should contact the manufacturer for more information. After the initial
quality control, it is advisable to schedule periodic quality control measurements, typically at
yearly intervals.
The quality control measurements discussed above are a necessity rather than an option
and are essential if the specified accuracy should be realized in everyday measurements. But
when there is a need to measure with an instrument more accurately, than what is its accuracy
specified by its manufacturer, there is the second reason why to calibrate the instrument. The
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accuracy of the instrument can be improved through an extensive calibration and improved
field procedures. Sometimes the proper calibration can be same efficient as buying a new instrument which has better accuracy given by the manufacturer.
3.2 Errors of instruments
3.2.1 Overview of errors
All electronic distance meters suffer from a large number of errors, irrespective of use of the
pulse measurement or the phase measurement principle. Errors are caused by electrical, optical and mechanical principles employed in instruments and by inaccurate manufacturing of
them. Magnitudes of these errors are kept small by manufacturers and they are accounted for
in the accuracy specifications of instruments. In view of the fact that errors occasionally exceed a specified accuracy and may change with time, users must be aware of unlimited trust in
their instruments.
Errors can be divided out to random and systematic, instrumental and non-instrumental,
periodic and non-periodic, distance-dependent and non-distance-dependent, linear and nonlinear, etc. Lists below demonstrate a number of instrumental and non-instrumental errors.
Instrumental errors are for example:

additive constant,

electrical and optical cross-talk,

multipath error,

resolution of pulse or phase measurement,

phase inhomogeneity and phase drift,

frequency errors,

error in given carrier wavelength,

effect of signal strength.
Non-instrumental errors are for example:

centering error of reflector,

leveling error of reflector,
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
reflector pointing,

measurement of temperature, pressure and humidity,

unwanted reflections from objects other than reflector,

atmospheric turbulences.
The aim of this section is not to give examination of each error separately. The goal for a user
is to understand consequences of errors which can be eliminated by calibration process.
3.2.2 The zero error and the additive constant
Because the virtual electro-optical origin or the center of an EDM instrument is usually not
located on the vertical axis of the instrument, a small correction has to be added to all distance
measurements to refer the distance to the instrument’s vertical axis. This correction is usually
called the additive constant and it compensates the dislocation between mechanical and electro-optical centers of the EDM instrument as well as changes of wavelength and velocity of a
signal inside the instrument. Similar disproportion between mechanical and optical centers
can be found on reflecting prisms. Therefore the additive constant usually summarises effects
of all these errors together.
Sometimes a term “the zero error” can be used to express similar value as the additive
constant. In fact, the zero error is more general term and it is used not only in the EDM field.
But there is an important difference. The zero error has an opposite mathematical sign than
the additive constant, which means it is a negative value. The additive constant has to be added to a measured distance to obtain the corrected distance but the zero error has to be subtracted from it. The value of the additive constant is originally determined by manufacturers
of EDM devices and reflectors and it can amount to several centimeters.
The original additive constant is accurate enough for many applications. But the residual
additive constant can change with time and it should be determined periodically and applied
to all measured distances by either changing the original correction or by adding in postprocessing. Test measurements have shown that the additive constant may also be affected by
a number of other parameters. The constant of most distance meters is temperature-dependent.
Some instruments were found to show voltage-dependent and signal-strength-dependent additive constants. So it means sometimes dependency between additive constant and measured
distance can be discovered, but it is not the main source of errors which should be deeply examined.
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3.2.3 The scale error and the scale correction
Scale errors in EDM instruments are caused primarily by oscillators and by emitting and receiving diodes. Several other factors are contributing to final scale error. The main source of
scale error outside the instrument is an incorrect first velocity correction. This problem was
discussed in chapter 2. A relationship between the scale error and the scale correction is similar to relationship between the zero error and the additive constant. It means the difference is
only an opposite mathematical sign.
The most significant oscillator error is its dependency on temperature. Most short range
distance meters feature temperature-compensated crystal oscillators as time bases. The frequency versus temperature characteristic of quartz crystal oscillators can be modeled by a
third degree polynomial. Because of this phenomenon, short range distance meters are also
affected by warm-up effect. It means that an oscillator frequency is dependent on how long
the distance meter has been operating. During operation, the distance meter produce heat
which warms up the oscillator and causes a change of frequency. The short-term stability of
oscillators is usually very good but the long-term stability (ageing) may be up to 1 ppm per
year.
Diode errors may affect the scale of a distance in several ways. An actual carrier wavelength of the diode’s emission can be different from the nominal value. It causes error in
computation of the first velocity correction. The actual carrier wavelength can be temperaturedependent and effect the scale similar way as temperature-dependent oscillator. At last, the
linear component of the effect of phase inhomogeneity across the emitting and receiving diode cause scale error of the EDM instrument. Because of all these sources of errors, the scale
correction should be periodically determined in calibrations.
3.2.4 Other errors
The zero error and the scale error are the most important factors describing an accuracy of an
EDM instrument. Monitoring these errors and computation of corrections is an essential purpose of calibrations. Unfortunately not all possible errors can be expressed these ways. There
are so called short periodic errors and non-linear distance-dependence errors. Its magnitudes
have decreased in modern EDM instruments. Testing of instruments to these errors mostly
requires special equipment such as an uncommon design baseline, special conditions such as
indoor laboratories and special instruments such as a laser interferometer.
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Short periodic (cyclic) errors can be observed in EDM instruments based on the phase
measuring principle. Error with wavelength equivalent to the measuring unit length (first order cyclic error) and its harmonics (higher-order cyclic errors) are mainly cause by electrical
or optical cross-talks between the reference and measuring signals. Cyclic errors can be dependent on distance, signal strength, time and others. Some contributions to these errors can
be physically explained but certainly not all of them. The final cyclic error is a mixture of a
number of different errors. The manufacturers reduce the amplitude of these errors by rigorous electrical shielding, separation of optical channels and special anti-reflex coatings. When
numerically determined, the errors can be partly eliminated with introducing corresponding
corrections. Sine and cosine functions are used to analyze the pattern of errors.
All distance-dependent systematic errors, which are repeatable and reproducible but do
not fit above introduced error-classes, may be termed non-linear distance-dependent errors. It
includes long-periodic errors as well as non-periodic errors. These errors are most likely
caused by inhomogeneities in signals. The reflector moving further and further away from the
EDM instrument will collect and return less and less of the signal causing loss of phase information and signal noise. Polynomial expressions of n-th degree are used to model the errors.
3.3 EDM calibration baselines
Ideally, an EDM instrument should be calibrated by mounting a reflector on a carriage and
moving it on a long straight rail. With the distance meter mounted at the end of the rail, many
distances should be measured all over the total working range and instantly compared with a
working standard of length. This measurement should be performed in constant atmospheric
conditions and it should be repeated at equally spaced temperatures all over the temperature
range of the instrument. Whole test should be proceeded periodically to analyze changes of
corrections in time. Since this ideal method is technically extremely difficult and would be
extremely expensive and prohibitive, simpler and cheaper methods have been developed to
obtain results of reasonable quality for corresponding amount of money and time.
It is needless to say that the concept described above can be applied for close distances in
laboratory conditions, at one temperature only and with laser interferometer as a length stand-
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ard, but unfortunately this is not the complex test of all instrumental errors although it is very
precise. As it is generally not practical to test instrumental corrections simultaneously against
all variables, separate tests are used to determine one coefficient or group of coefficients. For
obvious reasons, the ideal rail approach is usually replaced by a finite number of survey
marks (mostly pillars) along a line in outdoor. Such the test-line is typically called the EDM
calibration baseline. Calibrations on such test-lines are the main topic of this paper.
Field calibration baselines are the most common tools for performing standard calibrations of EDM instruments. They enable to find out the additive constant, the scale correction
and sometimes also the cyclic error of an EDM instrument. The additive constant can be determined without knowing baseline’s absolute distances but they have to be known when the
scale correction is computed. Special baseline design (Heerbrugg, Aarau, Hobart, etc.) with
regard to the instrument’s length unit is needed to analyze the cyclic error. Distances should
be evenly spread over the whole measuring range if possible. Extremely stable bottom layer
such se bedrock is required to ensure that the distances remain unaffected and stable. Ground
marks proofs good stability in time but it is complicated and inaccurate to center instruments
above them. On the other hand, pillars are suitable for accurate centering but they show shortterm movements (e.g. temperature, sun) as well as long term movements (e.g. season weather
changes). Combination of ground marks and pillars is sometimes applied to take advantages
of both stabilizations.
Measurement process can differ slightly or significantly dependent on a baseline, an
EDM instrument and required accuracy. But here are some notes which should be generally
followed:

all possible distances should be measured in both ways,

the EDM instrument station and the reflector station must be shaded,

temperature and air pressure are observed at both stations,

wet temperature or relative humidity is measured,

the EDM instrument should be tempered at least 15 minutes before measurement,

the additive constant is set to zero or employed reflector is chosen,

internal ppm correction (the first velocity correction) is set to zero,

fine mode of distance measurement should be set,

best resolution of distance reading should be chosen,

all distances are measured with the same reflector,
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
extra precise leveling of instrument and reflector is required,

at least four repetition of the measurement of each distance is demanded,

determining of heights of instrument and reflector above the pillars is needed.
3.4 Analysis of measurements
This section is not focus on pre-processing of measured data. After measurement, raw data
have to be prepared for the evaluation. Mostly important, the first velocity correction and geometrical correction are necessary to be applied to means of measured distances. This is descried step by step in section 4.4. Following sub-sections describe possibilities how to
evaluate the additive constant and the scale correction if there are true distances and preprocessed measured distances available.
3.4.1 Simple linear regression
In statistics, linear regression is an approach to modeling the relationship between a scalar
variable y and one or more explanatory variables denoted X. The case of one explanatory variable x is called simple regression and the case of more than one explanatory variable X is
called multiple regression. In linear regression, data are modeled using linear functions and
unknown model parameters are estimated from the measured data. Such models are called
linear models. Most commonly, linear regression refers to a model in which the mean of y at
the given value of x is an affine function of x. Simple linear regression, as well as regression
analysis in general, focuses on the probability distribution of y at given x. Values x are considered to be true values without errors. Analysis of joint probability distribution of y and x is
the domain of multivariate analysis. Linear regression models are often fitted using least
squares approach (ordinary least squares).
Summarized, simple linear regression is the least squares estimator of a linear regression
model with a single explanatory variable. In other words, simple linear regression fits a
straight line through the set of n points in such a way that makes the sum of squared residuals
of the model as small as possible. Residuals are called vertical distances between points of the
data set and the fitted line. The line can be expressed as follows (3.1).
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y    x
where
y
… dependent variable
x
… explanatory variable

… coefficient (intercept term)

… coefficient (slope term)
(3.1)
The sum of squares of residuals is as (3.2).
n
S  ,        xi  yi 
2
(3.2)
i 1
Partial derivations are set to be equal to zero for finding the minimum (3.3).
n
S
 2      xi  yi   0

i 1
n
S
 2      xi  yi   xi  0

i 1
(3.3)
By re-arranging equations, we can obtain (3.4).
n
n
n
  xi   xi2   xi yi
i 1
i 1
i 1
n
n
i 1
i 1
n
n
(3.4)
 n    xi   yi
And the final solution is as (3.5).
n

i 1
2
i
i 1
i
i 1
i
i 1
2
n
 n 
n   xi2    xi 
i 1
 i 1 
n

n
 x  y   x  x y
n
n
n   xi yi   xi   yi
i 1
i 1
i
i
(3.5)
i 1
2


n   xi2    xi 
i 1
 i 1 
n
n
Implementation of this statistical tool to calibration of EDM instruments is obvious. Values x represent baseline true distances and values y measured distances. Even more suitable is
to set y values as differences between true and measured distances (true - measured). Therefore α represents the additive constant and β represents the scale correction of calibrated EDM
instrument. Standard deviations of coefficients may be computed. A correlation coefficient,
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which can show amount of dependency between variables, is not very suitable for this application of simple linear regression. Much more interesting is calculation of the root mean
square deviation (RMSD) which indicates how observations are distributed along the regress
line.
n
RMSD 
where
r
… residuals
n
… number of residuals
r 
2
i
i 1
n
(3.6)
The simple linear regression is very sensitive to outliers. Even the only one false value in
the set of observations strongly affects the results. There are some advanced robust methods
how to deal with this problem. Robust methods take into account the residuals to weigh the
observations. But since there is a relatively small number of measurements in EDM calibrations, and outliers are not very frequented phenomena, it is not necessary to use robust methods. In addition, the results would be affected by some measurements more than others which
is not the right thing in most cases.
To use a regress line as a regression model is very advantageous. Coefficients of a line
can be used as instrumental corrections. But also more complex curve can be used as a regression model to fit the observed data better. Quadratic, cubic and higher-level polynomial lines
can be used for this purpose. RMSD error will slightly decrease, when higher polynomial is
used, but it become much more complicated to apply computed corrections. This is used
mostly in experimental applications. Methods, computing both instrumental corrections altogether, will be called “combined methods” later in this paper.
3.4.2 Separate computation
There is also a different way how to deal with an evaluation of measurements. Beside simple
linear regression, separate computation of the additive constant and the scale correction can
be applied. Pros and cons of both methods are discussed in following subsection 3.4.3. Socalled separate computation determines the additive constant using measured distances and
the least squares adjustment. The scale correction is then determined using corrected measured distances, baseline true distances and simple linear regression without the intercept term.
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In the first step, the additive constant is calculated by the least squares adjustment (LSA).
Input values for the adjustment are means of measured distances with applied corrections such
as the first velocity correction and the geometrical correction. Unknown variables are distances between pillars and the additive constant. An initial standard deviation of one measured
distance may be set according to accuracy specified by the manufacturer of the instrument.
All observations are usually equally weighted. No observations are rejected. The standard
LSA is then computed. As one of results, the additive constant is obtained. A standard deviation of the additive constant is then computed using a covariance matrix.
In the second step, the scale correction is evaluated using modified simple linear regression. The modification consists in omission of the intercept term. Modified regression model
looks like equation (3.7).
yx
(3.7)
Derivation of the final solution (3.10) is similar to ordinary simple linear regression.
n
S        xi  yi 
2
(3.8)
i 1
n
S
 2    xi  yi  xi  0

i 1
(3.9)
n

x
i 1
n
yi
i
x
i 1
(3.10)
2
i
RMSD and other statements introduced in subsection 3.4.1 are equally valid for simple linear
regression without the intercept term.
3.4.3 Comparison of methods
Simple linear regression may seem as the best way to evaluate measurements. It is wellknown, easy and it solves all together. But even if the estimate of corrections for particular set
of observations is the best possible with this method, there is a serious doubt if this computation is completely suitable for calibrations of EDM instruments. Employing simple linear regression in the evaluation process leads to calculating both corrections together without
respecting their initial definitions. The computed constants can be hardly called the additive
constant and the scale correction since each of them is a mixture of both. The contributions
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cannot be recognized in results and they cannot be separated. That makes impossible to trace
differences in corrections in time.
On the other hand, separate computation respects geodetic meaning of corrections. The
additive constant is computed separately using only measured distances. True baseline distances are not needed. Measured distances are corrected for computed additive constant and
then the scale correction is computed as simple linear regression without the intercept term.
That makes proper separation of constants which can be now called the additive constant and
the scale correction in a geodetic meaning. But this separate calculation does not give the best
possible results in a mathematic sense. There is always some small residual error which is not
covered by this computation and the constants do not fit for the particular set of measurements
as well as if they are evaluated using ordinary simple linear regression.
The difference in these two methods is not very large but it is significant. The difference
is not the same for a specific EDM instrument and it differs according to each set of measurements. The most interesting is the comparison of RMSD of both methods. RMSD of simple linear regression is always smaller than RMSD of separate computation but the difference
is insignificant in most cases. That is why the differences in corrected distances (resp. residuals) are insignificant using both methods even if the differences in constants may be significant. The difference in the additive constant is always largely compensated in the scale
correction. In spite of this small accuracy disadvantage, the separate computation has a big
advantage which is the traceability of corrections in time.
Section 3.4 uses bibliography: [2], [10], [16], [17], [18].
3.5 Uncertainty of measurement
3.5.1 Introduction
The term “uncertainty of measurement” was established to uniform the terminology. “The
uncertainty of measurement is a parameter, associated with the result of a measurement, that
characterizes the dispersion of the values that could reasonably be attributed to the measurand.” Quoted from [19]. The measurand is the particular quantity to be measured. The term
“uncertainty of measurement” is sometimes used only as “uncertainty” but it may have also
another meaning. In metrology, the term “error” does not mean the same thing as “uncertain-
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ty”. Uncertainty is a quantification of the doubt about the measurement result while error is
the difference between the measured value and the true value of the object being measured.
The importance of determining uncertainties should be obvious. No result can be considered
understandable without the associated uncertainty.
Errors and uncertainties come from very different sources which all have to be examined
and uncertainties estimated. In a measuring process, the sources may be such follows:

the measuring instruments (EDM instrument, reflector, thermometer, barometer, …),

the item being measured (calibration baseline),

the measurement process (measurement procedure),

imported uncertainties (projection correction, instrument calibrations),

operator skill (measurer),

the environment (atmospheric conditions),

sampling issues (atmospheric conditions determined at a wrong place).
Determining of uncertainties is very individual because it depends on many variables and
sometimes also on personal experience or a subjective attitude of the evaluator. Some of the
most important instructions related to uncertainty of measurement are presented in section
3.6. The most essential instruction in this field is the GUM [20] presented in subsection 3.6.2.
The simplified document, desired to be used by calibration laboratories, is the document EA
4/02 [19] presented in subsection 3.6.3. Further explanations and notations respect rather the
document EA 4/02 than the GUM if there are small variances between them.
In calibration, one usually deals with only one measurand or output quantity Y (corrected
distance) that depends upon a number of input quantities
(distance, temperature, pres-
sure, …) according to the function relationship (3.11).
Y  f  X1 , X 2 , ..., X n 
The model function
(3.11)
represents the procedure of the measurement and the method of evalu-
ation. It describes how values of the output quantity are obtained from values of the input
quantities. The set of input quantities may be grouped into two categories according to the
way in which the value of the quantity and its associated uncertainty have been determined:

Quantities whose estimate and associated uncertainty are directly determined in the
current measurement.
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Quantities whose estimate and associated uncertainty are brought into the measurement from external sources.
For a random variable, the variance of its distribution or the positive square root of the
variance, called standard deviation, is used as a measure of the dispersion of values. “The
standard uncertainty of measurement” associated with output estimate or measurement result
, denoted by
the estimates
, is the standard deviation of the measurand . It is to be determined from
of the input quantities
and their associated standard uncertainties
.
3.5.2 Type A evaluation of standard uncertainty
The type A evaluation of standard uncertainty can be applied when several independent observations have been made for one of the input quantities under the same conditions of measurement. If there is a sufficient resolution in the measurement process, there will be an
observable scatter or spread in the values obtained.
Assume that the repeatedly measured input quantity
the quantity
is the arithmetic mean ̅ (3.12) of
q
is the quantity . The estimate of
individual observed values
1 n
qj
n j 1
.
(3.12)
An estimate of the variance of the underlying probability distribution is the experimental variance that is given by (3.13).
s2  q  
2
1 n
qj  q 


n  1 j 1
(3.13)
Its positive square root is termed “the experimental standard deviation”. The best estimate of
the variance of the arithmetic mean ̅ is “the experimental variance of the mean” given by
(3.14).
s2  q 
s q  
n
2
(3.14)
Its positive square root is termed “experimental standard deviation of the mean”. The standard
uncertainty
̅ associated with the input estimate ̅ is the experimental standard deviation
of the mean ̅ (3.15).
u q   s q 
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When a number of
of repeated measurements is low, the reliability of a type A evalu-
ation of standard uncertainty has to be considered. If the number of observations cannot be
increased, other methods have to be taken into account. “Pooled estimate of variance”
based on a long statistical control or experience may be available. This pool estimate of variance should characterize the dispersion better than the estimated standard deviation obtained
from a limited number of observations. The variance of the mean may be estimated by (3.16)
and the standard uncertainty of the mean is deduced from (3.15).
s2  q  
s 2p
(3.16)
n
3.5.3 Type B evaluation of standard uncertainty
The type B evaluation of standard uncertainty is the evaluation of the uncertainty associated
with an estimate
of an input quantity
by means other than the statistical analysis of a
series of observations. The standard uncertainty
is evaluated by scientific judgment
based on all available information on the possible variability of
. Values belonging to this
category may be derived from:

previous measurement data,

experience with or general knowledge of the behavior of instruments and materials,

manufacturer’s specifications,

data provided in calibration and other certificates,

uncertainties assigned to reference data taken from handbooks.
The proper use of the available information for a type B evaluation of standard uncertainty of
measurement calls for insight based experience and general knowledge. It is a skill that can be
learned with practice.
3.5.4 Standard uncertainty of the output estimate
In the GUM [20], the standard uncertainty of the output estimate, obtained by combining type
A and type B evaluation of standard uncertainty, is defined as “combined standard uncertainty”. In the EA 4/02 [19], there is no mention about combined standard uncertainty since both
types of uncertainties are considered to be equally valid and more suitable one is always
picked to be used for the particular input quantity
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For uncorrelated input quantities the square of the standard uncertainty associated with
the output estimate resp. the measurement result
is the sum of squares given by (3.17).
N
u 2  y    ui2  y 
(3.17)
i 1
The standard uncertainty
the output estimate
estimate
is the contribution to the standard uncertainty associated with
resulting from the standard uncertainty associated with the input
according to (3.18).
ui  y   ci u  xi 
where
ci
(3.18)
… sensitivity coefficient
The sensitivity coefficient is the partial derivative of the model function
evaluated at the input estimates
with respect to
(3.19). It describes the extent to which the output estimate
is influenced by variations of the input estimate
ci 
f
f

 xi  X i
.
(3.19)
X1  x1 ... X N  xN
The uncertainty analysis for a measurement, sometimes called the uncertainty budget of
the measurement, should include a list of all sources of uncertainty together with the associated standard uncertainties of measurement and the methods of evaluating them. It is recommended to present the data relevant to this analysis in the form of a table containing:

quantity

type of evaluation of standard uncertainty,

associated standard uncertainty of measurement

the sensitivity coefficient

uncertainty contribution
, referenced by a physical symbol,
,
,
.
The dimension of each of the quantities should be also stated with the numerical values in the
table. In the bottom right corner of the table, the standard uncertainty associated with the
measurement result is given.
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3.5.6 Expanded uncertainty of measurement
Within EAL (Evaluation Uncertainty Assurance) it has been decided that calibration laboratories accredited by members of the EAL should state an “expanded uncertainty of measurement”, denoted by U. The expanded uncertainty is obtained by multiplying the standard
uncertainty
by a “coverage factor” k (3.20).
U  k u  y
(3.20)
In cases where a normal (Gaussian) distribution can be attributed to the measurand and the
standard uncertainty associated with the output estimate has sufficient reliability, the standard
coverage factor
2 shall be used. The assigned expanded uncertainty corresponds to a cov-
erage probability of approximately 95 %. If there is a reasonable doubt about reliability of the
standard uncertainty associated with output estimate see the document EA 4/02 for more details [19].
In calibration certificates, the complete result of the measurement consisting of the estimate y of the measurand and the associated expanded uncertainty U should be given in the
form ( y ± U ). To this an explanatory note must be added which in the general case should
have the following content:
The reported expanded uncertainty of measurement is stated as the standard uncertainty of
measurement multiplied by the coverage factor
2, which for a normal distribution corre-
sponds to a coverage probability of approximately 95 %. The standard uncertainty of measurement has been determined in accordance with EAL Publication EAL-R2.
Section 3.5 uses bibliography: [19], [20], [21].
3.6 Instructions related to EDM calibrations
This section is not meant to be a detailed description of legal instructions related to EDM calibrations. Such a topic is a very wide one and it is not the main focus of this paper. Reading of
a specific instruction is always unavoidable for proper understanding. This overview is giving
brief outlook in questions of legal background of calibrations of EDM instruments.
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3.6.1 Standards ISO 17123
Standards ISO 17123 bear common title “Optics and optical instruments – Field procedures
for testing geodetic and surveying instruments.” There are eight parts of the ISO 17123 standards.

Part 1: Theory

Part 2: Levels

Part 3: Theodolites

Part 4: Electro-optical distance meters (EDM instruments) [22]

Part 5: Electronic tachometers

Part 6: Rotating lasers

Part 7: Optical plumbing instruments

Part 8: GNSS field measurement systems in real-time kinematic (RTK)
“Standards ISO 17123 specify only field test procedures for geodetic instruments without
ensuring traceability in accordance with ISO/IEC Guide 99. For the purpose of ensuring
traceability, it is intended that the instrument be calibrated in the testing laboratory in advance.” Quoted from [23]. Standards are primarily intended to be used in engineering geodesy
as the control tool for checking suitability of instruments for intended use. Test procedures are
designed to be applied without special additional equipment and to minimize atmospheric
influences. Statistical tests for evaluating significance of results are present.
Part 1 and part 4 are those most related to the topic of calibrations of EDM instruments.
“ISO 17123-1 gives guidance to provide general rules for evaluating and expressing uncertainty in measurement for use in the specifications of the test procedures described in following parts.” Quoted from [23]. The following part 4 is dealing with testing of electronic
distance meters. Two different testing procedures are described – simplified test and full test.
A fundamental limitation of both of these procedures is a fact that they do not allow to detect
additional instrumental corrections. Accuracy and precision are tested as a complex feature of
an instrument without the aim of determining the additive constant and the scale correction.
Therefore the tests are not calibrations since no corrections can be obtained. Only additive
constant can be determined using full test but the main aim is still to determine experimental
standard deviation.
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The simplified test works with reduced amount of measurements. The purpose is to find
out if differences between true and measured distances are inside an accepted interval determined by
according to a standard ISO 4463-1. Calibration field or another precise EDM
instrument is required to run this test. From certain station, distances to other four points are
measured three times. Points are distant from about 20 to about 200 meters and they do not
have to be formed in a line. Mean distances and differences (true - measured) are computed. If
differences do not fit into the interval
, analyses of results must be performed. If there is
systematic error, additional instrumental corrections should be checked in certified calibration
laboratory. If there is not an obvious systematic error, full test procedure is recommended to
be carried out.
Full test should provide more accurate estimation of the experimental standard deviation
of a tested instrument. Distances between all combinations of seven points in a line of 600
meters are measured in forward direction only. True distances are not needed and remain unknown in this test therefore no information about the scale can be derived. Seven points of a
line are placed according to a carrier wavelength of the tested instrument. Calculation is based
on a simple least squares adjustment with distances and an additive constant as unknown variables. Adjusted distances and corresponding standard deviations are not evaluated. The experimental standard deviation of the instrument is determined using residuals of all measured
distances.
s
where
r
… residuals
v
… degrees of freedom
r
2
v
(3.21)
3.6.2 Guide to the expression of uncertainty in measurement
The GUM (Guide to the expression of uncertainty in measurement) is an essential document
dealing with uncertainties of measurement. Guide was written in 1995 with minor corrections
in 1998 by the JCGM (Joint Committee for Guides in Metrology) member organizations
(BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML). Full name of the document is
“Evaluation of measurement data – Guide to the expression of uncertainty in measurement.”
In 2009, JCGM published related publication named “Evaluation of measurement data – An
introduction to the Guide to the expression of uncertainty in measurement and related docu-
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ments”. The GUM is freely available on the BIMP website [20]. Section 3.5 respects this
guide.
The aim was: “To develop a guidance document based upon the recommendation of the
BIPM Working Group on the Statement of Uncertainties which provides rules on the expression of measurement uncertainty for use within standardization, calibration, laboratory accreditation, and metrology services. The purpose of such guidance is to promote full
information on how uncertainty statements are arrived at and to provide a basis for the international comparison of measurement results.” Quoted from [20].
3.6.3 Document EA 4/02
This document is a publication of the EA (European co-operation for Accreditation). Full
name of the document is “Expression of the Uncertainty of Measurement in Calibration”. This
document is freely available on websites of the EA [19]. Section 3.5 respects this document.
“The purpose of this document is to harmonize evaluation of uncertainty of measurement
within EA, to set up, in addition to the general requirements of EAL-R1, the specific demands in
reporting uncertainty of measurement on calibration certificates issued by accredited laboratories
and to assist accreditation bodies with a coherent assignment of best measurement capability to
calibration laboratories accredited by them. As the rules laid down in this document are in compliance with the recommendations of the Guide to the Expression of Uncertainty in Measurement,
published by seven international organizations concerned with standardization and metrology,
the implementation of EA-4/02 will also foster the global acceptance of European results of
measurement.” Quoted from [19].
In first parts of the document, terms such as standard uncertainties type A and B and expanded uncertainty are defined. These parts are very similar to GUM. In following appendixes, more terms are explained. In the most extensive part, there are supplements with examples
of calibrations of several different instruments. Even if the document is very useful reading to
familiarize the problematic of uncertainties of measurements, there is no suitable example of
calibrating an EDM instrument. In spite of the document should be practical example of how
to evaluate uncertainties of measurement, it cannot cover the whole range of instruments and
calibration specifics. In these cases, contacting the EA organization is advised.
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3.6.5 Document M3003
Document M3003, named “The Expression of Uncertainty and Confidence in Measurement”,
is published by the UKAS (United Kingdom Accreditation Service). Second edition is freely
available since January 2007 on websites of the UKAS [24]. Same as the document EA 4/02,
the M3003 is based on information in the GUM.
“The purpose of these guidelines is to provide policy on the evaluation and reporting of
measurement uncertainty for testing and calibration laboratories. Related topics, such as
evaluation of compliance with specifications, are also included. A number of worked examples are included in order to illustrate how practical implementation of the principles involved can be achieved.” Quoted from [24].
3.6.6 Standard ISO/IEC 17025
International standard ISO/IEC 17025:2005 named “General requirements for the competence
of testing and calibration laboratories” was developed by organizations IAF, ILAC and ISO.
Unless the instruction is not intended to be used as the basis for certification of laboratories, it
is sometimes used as one of requirements. The certification process is very expensive and
therefore it is adopted only by professional laboratories. The standard is not freely available
and was not owned by the author while writing this paper.
“ISO/IEC 17025:2005 specifies the general requirements for the competence to carry out
tests and/or calibrations, including sampling. It covers testing and calibration performed using standard methods, non-standard methods, and laboratory-developed methods. It is applicable to all organizations performing tests and/or calibrations regardless of the number of
personnel or the extent of the scope of testing and/or calibration activities. When a laboratory
does not undertake one or more of the activities covered by ISO/IEC 17025:2005, such as
sampling and the design/development of new methods, the requirements of those clauses do
not apply. ISO/IEC 17025:2005 is for use by laboratories in developing their management
system for quality, administrative and technical operations. Laboratory customers, regulatory
authorities and accreditation bodies may also use it in confirming or recognizing the competence of laboratories.” Quoted from [25].
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3.6.8 Standard ISO 10012
International standard ISO 10012:2003 named “Measurement management systems - Requirements for processes and measuring equipment” is the revised standard of ISO 100121:1992 and ISO 10012-2:1997. ISO 10012:2003 is not intended as a substitute for or as an
addition to the requirements of ISO/IEC 17025. And it is not intended to be used as a requisite
for demonstrating conformance with ISO 9001. Interested parties can agree to use ISO
10012:2003 as an input for satisfying measurement management system requirements in certification activities. It can be used to create a metrological confirmation system in a company.
The standard is not freely available and was not owned by the author while writing this paper.
“ISO 10012:2003 specifies generic requirements and provides guidance for the management of measurement processes and metrological confirmation of measuring equipment used
to support and demonstrate compliance with metrological requirements. It specifies quality
management requirements of a measurement management system that can be used by an organization performing measurements as part of the overall management system, and to ensure metrological requirements are met.” Quoted from [26].
3.6.9 Standards ISO 9000
The ISO 9000 family of standards represents an international consensus on good quality management practices. It consists of standards and guidelines relating to quality management systems and related supporting standards. The ISO 9000 family of standards is named “Quality
management systems” and there are three main parts.

ISO 9000:2005 – Fundamentals and vocabulary

ISO 9001:2008 – Requirements

ISO 9004:2009 – A quality management approach
ISO 9001:2008 is implemented by over a million organizations in 176 countries. All requirements are generic and are intended to be applicable to all organizations, regardless of
type, size and product provided. Where any requirement(s) of ISO 9001:2008 cannot be applied due to the nature of an organization and its product, this can be considered for exclusion.
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“ISO 9001:2008 specifies requirements for a quality management system where an organization:

needs to demonstrate its ability to consistently provide product that meets customer
and applicable statutory and regulatory requirements,

aims to enhance customer satisfaction through the effective application of the system,
including processes for continual improvement of the system and the assurance of
conformity to customer and applicable statutory and regulatory requirements.
Quoted from [27].
Quality management systems are also concern calibration laboratories as well as geodetic
companies. Anyone who would like to proof that the quality management system of a company is in conformity with ISO 9000 family of standards has to pass through a certification process organized by an independent certification body. The certification is very expensive and
following paid audits are necessary. Therefore every company must calculate pros and cons of
ISO 9000 certification. The standards are not freely available and were not owned by the author while writing this paper.
Section 3.6 uses bibliography: [2], [19], [20], [21], [22], [23], [24], [25], [26], [27].
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4. Calibration at the Nummela baseline
4. CALIBRATION AT THE NUMMELA BASELINE
4.1 The Nummela Standard Baseline
4.1.1 The Finnish Geodetic Institute
The Finnish Geodetic Institute, as the name predicates, is an organization dealing with geodesy. It is a research institute of the Ministry of Agriculture and Forestry of Finland specializing
in geodesy and geospatial information, science and technology. It was founded in the year
1918, its residence is situated in Masala, twenty-five kilometers west of Helsinky. The FGI
has personnel of about eighty people in five departments:

Geodesy and Geodynamics,

Geoinformatics and Cartography,

Remote Sensing and Photogrammetry,

Navigation and Positioning,

Administration Services.
Department of Geodesy and Geodynamics is concerned with reference frames, height
systems, geoid models, crustal deformation and metrology. It operates the Metsähovi Fundamental Geodetic Station with advanced measuring methods such as: VLBI, SLR, GNSS,
DORIS, AG, SCG. This department also includes National Standards Laboratory of Acceleration of Free Fall and National Standards Laboratory of Length. National Standards Laboratory
of Length deals with calibration of geodetic instruments such as electronic distance meters
and leveling rods. It operates calibration laboratory for precise-leveling and digital leveling
systems, test field for leveling and Nummela baselines (the Nummela Standard Baseline, the
Nummela Calibration Baseline). The Nummela Calibration Baseline is derived from the
Nummela Standard Baseline using high-precision EDM instruments and it is opened to public.
Subsection 4.1.1 uses bibliography: [28].
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4.1.2 Baseline description
The Nummela Standard Baseline of the FGI is a unique national and international measurement standard for length measurements in geodesy. In 1933 a new baseline was established in
Nummela, 45 km NW of Helsinki, on a frost-resistant ridge of moraine and sand with glacial
origin covered with pine forest. The design was fitted for comparison of invar wires (36 x 24
m = 864 m). Later, the baseline was equipped for the Väisälä interference comparator and
since the first interference measurements in 1947 it has been called the Nummela Standard
Baseline. All observation pillars are placed on the same line in space. The height difference
from 0 m to 864 m is -4.1 m. Today it consists of 6 benchmark bolts in underground concrete
blocks at 0, 24, 72, 216, 432 and 864 meters which are generally called underground markers
(Fig. 9). 6 observation pillars (Fig. 10) with Kern-type forced-centering are situated alongside
underground markers. Unique dismountable Kern centering plates completes the centering
system (Fig. 11) for purpose of fixing standard geodetic equipment. Aboveground pillar constructions were recently reconditioned in 2007.
Fig. 9: The underground marker at 24 m
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Fig. 10: The observation pillars at 0 m (left) and 864 m (right)
Fig. 11: The Kern-type forced-centering system
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The traceability from the definition of the meter to a distance of the baseline is provided
through the quartz gauge system and measurements with the Väisälä interference comparator.
Importance of Väisälä standard baselines was acknowledged already in 1951 by the IAG
General Assembly and in 1954 by the IUGG, which recommended such measurements in
different countries all around the world. At the Nummela Standard Baseline, 15 interference
measurements during 1947-2007 verified traceability and extreme stability of the baseline.
For the length of 864 m, all results during 60 years are within 0.6 mm, latest total standard
uncertainty for this distance is
0.08 mm. Results of interference measurements is preserved
in length between underground markers, which can be accessed and utilized through regular
theodolite-based projection measurements and observation pillars.
Subsection 4.1.2 uses bibliography: [8], [29], [30], [31].
4.1.3 Quartz gauge system
A measurement of a baseline with the Väisälä interferometer is a high accuracy length measurement in geodesy. An accurate length up to 1 km is achievable by interferometrically multiplying the length of a 1 m quartz bar. Therefore, the absolute length of the bar should be
known with small uncertainty. A measurement setup and procedure for absolute length calibration of a quartz bar by combined white and laser light gauge block interferometer has been
developed. A standard uncertainty of 35 nm has been achieved with these methods by Tuorla
Observatory. The system of tens of quartz meters (Fig. 12) has been maintained for more than
70 years. The lengths are determined through repeated comparisons and absolute calibrations.
The most commonly used quartz gauges are 1 m long, 23 mm thick hollow quartz tubes
sealed with 10 to 15 mm thick cylindrical ends. The bars have hollow core but monolithic
ends. Quartz, as the material for the bars, was selected due to its low thermal expansion coefficient. On the other hand, relatively large pressure coefficient and modulus of elasticity have
been found. Repeated measurements are necessary since the lengths of quartz gauges change
slightly over time. Slow lengthening of most bars has been observed. The present principal
normal of the quartz gauge system is the bar no. 29. Regular absolute calibration with white
and laser light gauge block interferometer of this bar brings scale to relative comparisons (Fig.
12) with other quartz meters.
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Fig. 12: Quartz gauges (left) and quartz gauge comparator (right) at Tuorla Observatory
With the present constellation of observations pillars, the only quartz gauge that can be
used in the Väisälä interference comparator at the Nummela Standard Baseline is the exceptionally long quartz gauge no. VIII. For example, a 100 μm shorter bar would produce an 86.4
mm shorter baseline, demanding modified positions of observation pillars. The length of
quartz gauge no. VIII is always determined before and after interference measurements at
Nummela (Fig. 13). Several corrections, especially from temperature, have to be applied to
achieve the absolute length under standard conditions. The final length may be the result of
latest comparisons as well as the result of calculations based on the long time series of observations. Usually, the final length is the average of both. During each interference measurement at the Nummela Standard Baseline, the actual length of quartz gauge no. VIII is
computed by correcting the standard length with temperature and atmospheric pressure corrections.
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Fig. 13: Comparisons of the length of quartz gauge no. VIII at Tuorla Observatory
Note: The black spot at 2000 signifies the absolute calibration of this particular gauge at MIKES.
4.1.4 Interference measurements
The Nummela Standard Baseline is calibrated every few years through interference measurements with the Väisälä interference comparator. The FGI is currently the only institute, which
carries out these high-precision measurements, latest in 2005 and 2007. Next observation is
scheduled to 2013. The measurements are based on multiplication of length of a 1-m-long
quartz gauge. A longer length is always a multiple of a shorter length. At the Nummela
Standard Baseline, 864 m length is realized by using several multiplications: 2 x 2 x 3 x 3 x 4
x 6 x 1 m = 864 m. The method is basically simple, but extremely laborious to put into practice. Installation of the Väisälä interference comparator at the Nummela Standard Baseline
takes about two weeks.
The main parts of the Väisälä interference comparator (Fig. 14, Fig. 15, Fig. 16), mostly
iron and glass, must be measured and fixed at proper locations with ±1 mm 3D-accuracy. The
rest is controlled with the numerous adjusting screws around the mirrors in the comparator. A
quartz meter (currently the no. VIII) brings the scale to the system. White light from a pointlike source is used. The beam is made parallel with a collimator lens and divided into two
beams. One part of the light travels between the front mirror and the middle mirror, the other
part travels to and from the back mirror (Fig. 14). The distance between the front and back
mirrors is an integer multiple n of the distance between the front and middle mirrors. The light
beam travels n times between the first two mirrors and once to and from the back mirror. The
mirrors are adjusted in such a way that the two beams, travelling different paths, but equal
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distances, meet at the focal plane of the observing telescope. The light source and the telescope include numerous fine-mechanical and optical components to control the light beams.
The final adjustment of the incoming beams with adequate accuracy is made with the screen
and the compensator glasses by delaying one of the two beams in front of the telescope.
Fig. 14: The principle of the Väisälä interference comparator (interference 0-6-24)
Fig. 15: Instruments of the Väisälä interference comparator at the pillars 0 and 1
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Fig. 16: The mirror equipment of the Väisälä interference comparator
Interference observations require extremely stable temperature conditions for length of
hundred meters and longer. Otherwise it is not possible to control the course of reflecting
beams and to adjust the mirrors and compensators with the required 0.001 mm accuracy to
find interference fringes. For a one successful set of measurements, small (≤ 1° C) temperature differences are needed not only along the baseline but also during a seven hour observation period. In Nummela, during a two-three months autumnal measurement period, only a
few cloudy nights are usually suitable for observation. In stable conditions, modeling the refraction is not a problem, since only the refraction difference between the divided two beams
is needed. 31 thermometers are placed along the baseline to monitor these differences. Two
observers and two assistants are needed almost seven days and nights a week during a measurement period.
When the quartz gauge is placed between mirrors at 0 and 1 m, 0-1-6 interference may be
observed and the length 0-6 could be determined. The 6-m-interference is then multiplied to
longer length. The positions of mirrors are registered with a special transferring device relative to permanent transferring bars on every pillar and projected later between underground
markers for further use. Projection measurements are performed before and after the interference measurement. This repetition reduces the total uncertainty, in which the projection
measurements are a major factor, together with the uncertainty of the quartz meter length.
Reverse projection correction are needed for EDM calibrations at observation pillars. In the
year 2007 projections caused from 0,016 to 0,033 mm standard uncertainty, random errors in
interference observations and transfer readings caused from 0,010 to 0,054 mm standard un-
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certainty. The length of the quartz bar caused from 0,001 to 0,030 mm uncertainty in baseline
lengths. Other smaller type B uncertainties were evaluated on the basis of experience with
previous measurements during 60 years.
Lengths of full time series of observations with total standard uncertainties are listed in
(Table 2). They are lengths between underground benchmarks, reduced to the height level of
underground marker at 0 m. Data from the first three shortest lengths during the years 19471975 are missing due to the absence of underground markers and observation pillars at these
distances. Data from the 864-m-length from 1983 and 2005 are missing due to unfavorable
weather conditions in which the measurements cannot be done. Latest measurements in 2005
and 2007 are in congruence with previous results and once again they proofed the stability of
the baseline, great repeatability of observations and small uncertainties of measurement.
Table 2: The Nummela baseline lengths during the years 1947-2007
Epoch
1947.7
1952.8
1955.4
1958.8
1961.8
1966.8
1968.8
1975.9
1977.8
1983.8
1984.8
1991.8
1996.9
2005.8
2007.8
The Nummela baseline lengths and standard uncertainties
0 - 24
0 - 72
0 - 216
0 - 432
0 - 864
[mm + 24 m] [mm + 72 m] [mm + 216 m] [mm + 432 m] [mm + 864 m]
------95.46 ± 0.04 122.78 ± 0.07
------95.39 ± 0.05 122.47 ± 0.08
------95.31 ± 0.05 122.41 ± 0.09
------95.19 ± 0.04 122.25 ± 0.08
------95.21 ± 0.04 122.33 ± 0.08
------95.16 ± 0.04 122.31 ± 0.06
------95.18 ± 0.04 122.37 ± 0.07
------94.94 ± 0.04 122.33 ± 0.07
33.28 ± 0.02 15.78 ± 0.02 54.31 ± 0.02 95.10 ± 0.05 122.70 ± 0.08
33.50 ± 0.02 15.16 ± 0.02 53.66 ± 0.04 95.03 ± 0.06
--33.29 ± 0.03 15.01 ± 0.03 53.58 ± 0.05 94.93 ± 0.06 122.40 ± 0.09
33.36 ± 0.04 14.88 ± 0.04 53.24 ± 0.06 95.02 ± 0.05 122.32 ± 0.08
33.41 ± 0.03 14.87 ± 0.04 53.21 ± 0.04 95.23 ± 0.04 122.75 ± 0.07
33.23 ± 0.04 14.98 ± 0.04 53.20 ± 0.04 95.36 ± 0.05
--33.22 ± 0.03 14.95 ± 0.02 53.13 ± 0.03 95.28 ± 0.04 122.86 ± 0.07
4.1.5 Projection measurements
In projection measurements, the temporary locations of the mirrors on observation pillars are
projected onto the line between underground markers. Projection measurements are theodo-
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lite-based high-precision measurements which are needed before and after, and usually also
during the long-lasting interference measurement period. Reverse projections are necessary
for EDM calibrations at the Nummela Standard Baseline. The lengths preserved between the
underground markers are restored to lengths between forced-centering plates on observation
pillars with repeated projection measurements. Usual frequency is three or four times a year
due to measureable pillar movements during seasons.
Projection measurements are based on precise horizontal angle measurements. A theodolite is adjusted on an observation pillar, pointings are made and angles are read to distant targets on one or two other observation pillars in the baseline direction and to a plumbing rod
that is adjusted above the corresponding underground marker perpendicular to the baseline
direction. For one projection, four sets of horizontal angles are measured in two theodolite
face positions. The distance between the observation pillar and the underground marker is
measured with a calibrated steel tape. Due to optimal geometry, one millimeter uncertainty of
the length is easily obtained and sufficiently accurate. The projection corrections are calculated with trigonometric formulas. Even if there is a systematic difference in projections obtained from two different theodolites, this variation is not corrected since it would not be valid
for other instruments. The result is calculated as an average of both. Empirically, based on
long experience, the standard uncertainty of projection corrections was estimated to be ±0.07
mm.
Fig. 17: A plumbing rod above an underground marker (left), a set of adaptor bars (right)
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4.1.6 Scale transfers to other baselines
The Väisälä-type Nummela Standard Baseline of the FGI is the state-of-the-art geodetic baseline in the world. The FGI participated in calibrations of baselines using the Väisälä interference comparator at several places all around the world. Considering shipping problems, the
amount of time and number of specialists necessary for measurements, this cannot be done
very often and regular measurements at each baseline is not possible. Therefore a simpler
method was developed by the FGI. Current most accurate EDM instruments are used for scale
transfers from the Nummela Standard Baseline to other baselines. Mostly often, the Kern Mekometer ME5000 of the FGI is used. It is calibrated at the Nummela Standard Baseline several times before and after the scale transfer to other baseline. Whereas a baseline, calibrated
with the Väisälä interference comparator, has the standard uncertainty about 0.1 mm/km, a
baseline, calibrated with Kern Mekometer ME5000, has the standard uncertainty about 0.3
mm/km. During the last 15 years the scale of the Nummela Standard Baseline has been transferred to nearly 20 baselines and test fields in more than 10 countries all around the world.
4.2 Instruments and equipment
4.2.1 The Leica TCA2003
The first calibrated EDM instrument is an electronic distance meter integrated in Leica
TCA2003 total station (Fig. 18). The instrument was calibrated together with a reflector Wild
Leitz GPH1P. The instrument is owned by Aalto University, School of Engineering, Department of surveying. Lending was arranged by prof. Martin Vermeer.
Leica TCA2003 is the high-performance total station intended to be used for precision
measurement. This specific instrument was manufactured in the year 2000. It was last calibrated on the 6th March 2001 by the Institute of Geodesy (HUT). Although the model
TCA2003 was produced since 1997 and so it is not the very modern instrument, it has become
well known because of its exceptional precision and accuracy and it has been used for many
calibrations of EDM baselines. Still nowadays it is one of the most precise electronic distance
meters.
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The manufacturer dedicates the accuracy of 1 mm + 1 ppm. The heard of the electronic
distance meter inside Leica TCA2003 is a stable quartz oscillator with a guaranteed drift less
than 1 ppm. It uses the phase measurement principle and the source of light is an invisible
coaxial infrared laser with a wavelength of 805 nm. Measurement of an individual distance in
the standard mode takes about 3 seconds. The distance range is about 5 kilometers. The reference refractive index is 1.000281800 which corresponds to reference conditions (temperature
12 °C, pressure 760 Torr, relative humidity 60 %). Working temperature range is from
°C to 50 °C under maximum relative humidity 95 % (non-condensing).
Fig. 18: Leica TCA2003 + Wild Leitz GPH1P
Table 3: Constants used for calculations – Leica TCA2003
Constants - Leica TCA2003
Quantity
Value
EDM carrier wavelength:
850
Reference temperature:
12
Reference atm. pressure:
1013.25
Reference relative humidity:
60
Reference refractive index:
1.00028180
Initial additive constant:
0.00
Initial scale correction:
0.00
EDM instrument height:
0.240
Reflector height:
0.240
Assmann psychrometer constant:
0.000662
Radius of the Earth:
6370000
75
Unit
nm
°C
hPa
%
mm
mm
m
m
m
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4.2.2 The Kern Mekometer ME5000
The second calibrated instrument is an electronic distance meter Kern Mekometer
ME5000 (Fig. 19). The instrument was calibrated together with a supplied reflector Kern
RMO5035. The instrument is owned by Aalto University, School of Engineering, Department
of surveying. Lending was arranged by prof. Martin Vermeer.
Kern Mekometer ME5000 is the precision electronic distance meter. It is the older instrument considering that the calibrated instrument was manufactured in the year 1994. But
still today, Mekometer ME5000 is known as the most precise medium range distance meter in
the world with accuracy of 0.2 mm + 0.2 ppm declared by the manufacturer. This type of instrument has become very valuable with time since it is no more produced and since no other
type of instrument with equivalent precision and accuracy is available on the market. The calibrated instrument is the only instrument of its kind in Finland. It is widely used by the FGI as
the length transfer standard between calibration baselines. The instrument is periodically calibrated at the Nummela Standard Baseline several times a year.
Distance measurement of Mekometer ME5000 is based on a modified phase measuring
principle. Rather than using fixed modulation frequencies and measuring phase differences,
the modulation frequency is adjusted within a certain range until the transmitted and received
signals are in phase. This is done at four frequencies, namely at both ends of the tuning range
and twice in the middle of it. All measured frequency differences are used to calculate a displayed distance. Known frequencies are obtained from a frequency synthesizer. The source of
light is a HeNe infrared laser with a wavelength of 632.8 nm. Measurement of an individual
distance in the standard mode takes about 2 minutes. The distance range is from 20 meters to
about 8 kilometers. The reference refractive index is 1.000284514844 which corresponds to
reference conditions (temperature 15 °C, pressure 760 Torr, relative humidity 0 %, CO2 concentration 0.03 %). Working temperature range is from
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Fig. 19: Kern Mekometer ME5000 + Kern RMO5035
Table 4: Constants used for calculations – Kern Mekometer ME5000
Constants - Kern ME5000
Quantity
Value
EDM carrier wavelength:
632.8
Reference temperature:
15
Reference atm. pressure:
1013.25
Reference relative humidity:
0
Reference refractive index: 1.000284514844
Initial additive constant:
0.00
Initial scale correction:
0.00
EDM instrument height:
0.330
Reflector height:
0.330
Assmann psychrometer constant:
0.000662
Radius of the Earth:
6370000
Unit
nm
°C
hPa
%
mm
mm
m
m
m
4.2.3 Other used instruments and equipment
The EDM instruments are not the only devices used in the calibration process. The same
attention should be given to instruments used for measuring atmospheric conditions. Last but
not least, the employed reflectors should be explored.
The first used reflector was “Wild Leitz GPH1P” (Fig. 20) owned by Aalto University,
School of Engineering, Department of Surveying. No serial number of the reflector was
found. Unique identification should be established by an owner if several same type reflectors
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are used in a company. There is a stick with a sign “TKK/M-OS” on used prism. It is a robust,
high precision manufactured prism in a metal holder. Metal holders usually proof better durability than plastic ones. The eccentricity, which means the difference between the optical and
the mechanical centers of the reflector, is presented by the manufacturer as 0.0 mm. The reflector was placed into a standard geodetic tribach manufactured by Wild.
Fig. 20: The reflector “Wild Leitz GPH1P”
The second used reflector was manufactured by Kern and denoted as “Kern RMO5035”
(Fig. 21). It is owned by Aalto University, Shool of Enfineering, Department of Surveying.
The serial number of the prism is 374414. There is a stick with a sign “HUT/GEODESY
M023” on the metal holder. The reflector is specially designed to be used with Kern Mekometer ME5000. The target for optical pointing is below the prism. The height difference between
prism and optical target reflects the height difference between the laser line and the line of
sight of Kern Mekometer ME5000. The difference between the optical and mechanical centers of the reflector is presented by the manufacturer as 0.0 mm. The reflector was placed into
a standard geodetic tribach manufactured by Leica.
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Fig. 21: The reflector “Kern RMO5035”
Two Assmann-type psychrometers “Thermometers Thies Clima“ (Fig. 22) were used to
meausere dry and wet temperatures. Instruments are owned by the FGI and they are
standardly used in calibrations. Destiled water is used to moisten the wet bulb. The smallest
segment of the scale is 0.2 °C but readings can be easily performed with resolution of 0.1 °C.
The temperature measuring range is from
35 °C to
40 °C. No corrections for temperatures
were added since previous calibration of psychrometers produced corrections less than 0.1 °C
with standard uncertainty
0.08 °C. Standard uncertainty of the measurement of temperatures
with two psychrometers is estimated by the FGI as 0.11 °C.
Fig. 22: Assmann-type psychrometer “Thermometer Thies Clima”
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Two aneroid barometers „Thomen altimeters“ (Fig. 23) were used to observe air pressure.
Instruments are owned by the FGI and they are standardly used in calibrations. The smallest
segment of the scale is 1 hPa but finer resolution can be obtained by estimating segments of
0.1 hPa. The aneroids are periodically compared with a mercury barometer of the FGI and
corrections are determined. A possible drift is monitored for a long period of time. Standard
uncertainty of calibration of barometers is 0.1 hPa. Standard uncertainty of air pressure observations with two barometers according to the FGI is 0.2 hPa.
Fig. 23: The aneroid barometers "Thomen altimeters"
Relative humidity was determined using digital hand-held humidity and temperature
meter “Vaisala HM 34“ (Fig. 24). The device was borrowed from Aalto University, School of
Engineering, Departement of Surveying. Measurement range is from 0 % to 100 % relative
humidity. Displayed resolution is 0.1 %. Manufacturer indicated value of accuracy is
between 0 % and 90 % relative humidity and
2%
3 % between 90 % and 100 % relative
humidity. For proper results, calibration of the instrument is required. Unfortunately no
information about previous calibrations were obtained because most likely there were non
calibrations of the device. Since this information is not available, the real accuracy of the
instruments remains unknown.
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Fig. 24: The relative humidity meter “Vaisala HM 34”
4.3 Measurement
4.3.1 Measurement information

calibration baseline:
Nummela Standard Baseline, Finland

calibrated instruments:
a) Leica TCA2003, S.No. 438743
+ reflector Wild Leitz GPH1P, S.No. Unknown
b) Kern Mekometer ME5000, S.No. 357094
+ reflector Kern RMO5035, S.No. 374414


dates of measurement:
weather:
2011-10-06
- Leica TCA2003
2011-10-07
- Leica TCA2003
2011-11-24
- Kern Mekometer ME5000
2011-11-25
2011-11-28
2011-10-06
- cloudy, rainy, light wind
2011-10-07
- partly cloudy, moderate wind
2011-11-24
- cloudy, rainy, moderate wind
2011-11-25
- cloudy, drizzly, calm
2011-11-28
- cloudy then clear, windy
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temperature observation: Thermometer Thies Clima, S.No. 1101459
Thermometer Thies Clima, S.No. 1101460

pressure observation:
Thomen altimetr, S.No. 126533
Thomen altimetr, S.No. 164610

humidity observation:
Vaisala HM 34, S.No. 312338

measurer:
Bc. Filip Dvořáček

helping surveyors:
Bc. Petra Stolbenková
Panu Salo
Table 5: Overview of measurements
Overview of measurements
Date of
Calibrated
measurement instrument
2011-10-06
2011-10-07
2011-11-24
2011-11-25
2011-11-28
Leica
TCA2003
Kern
ME5000
Calibration status
Measured pillars
Completed 1st
0, 24, 72, 216, 432, 864
nd
0, 24, 72, 216, 432, 864
Completed 2
Incomplete 1st
st
Completed 1
0, 24, 72, 216
nd
Incomplete 2
432, 864
Completed 2nd
0, 24, 72
216, 432, 864
Table 6: Ranges of atmospheric conditions during calibrations
Ranges of atmospheric conditions during calibrations
Temperature [°C] Pressure [hPa] Rel. humidity [%]
Calibrated
Calibration
instrument
Min
Max
Min
Max
Min
Max
Leica
TCA2003
Kern ME5000
1st
12.5
13.2
976.9
979.2
97
99
2nd
8.6
11.8
974.2
976.2
58
85
st
1
5.6
7.4
994.3
999.7
82
99
2nd
1.5
8.2
975.9
993.0
73
98
4.3.2 Measurement procedure
The FGI measurement process was adopted in performed calibrations. It meets all requirements and principles for achieving the best possible results of calibrations in field conditions
under reasonable amount of time. The measurement process was demonstrated by Lic. Sc.
Jorma Jokela, the responsible person for length metrology of the FGI. When the FGI performs
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calibrations for private subjects, similar measurement process is used. Observed data were
recorded to small notebooks (Appendix 1, Appendix 2), no prearranged forms were used.
Fig. 25: Calibrating the Leica TCA2003 + the Wild Leitz GPH1P
Fig. 26: Calibrating the Kern Mekometer ME5000 + the Kern RMO5035
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Distances between all combinations of two pillars were observed. It means 5 different
distances from 6 different stations were measured. It is 30 distances together. The standard
order of measurements was as shown in (Table 7). Exceptions were possible if there were
reasons for them.
Table 7: Order of measured distances
EDM station order
Reflector station order
0
24 72 216 432 864
24
864 432 216 72
0
72
0
24 216 432 864
216
864 432 72 24
0
432
0
24 72 216 864
864
432 216 72 24
0
The instrument and the reflector were centered on pillars by screwing tribrachs to Kern
force-centering plates and they were leveled afterwards. The reflector was directed to the
EDM instrument using mechanical collimator. Every specific distance was measured 5 times
(Leica TCA2003) resp. 2 times (Kern Mekometer ME5000). Kern Mekometer ME5000
measures one distance about 2 minutes and the result itself is the mean of several partial results, therefore 2 times measurement is sufficient in most cases. New pointing to the target
was carried out before each measurement to make measurements more independent and eliminate systematic error caused by wrong pointing. Pointing was done optically (Leica
TCA2003) resp. electronically (Kern Mekometer ME5000). In this case, electronic pointing
means manual adjusting of the laser-line to find the maximum reflection and the maximal
strength of a returned signal. Displayed values were recorder with a resolution of 0.1 mm.
Variations in measured distances were monitored, but no values were unused. When there was
an obvious trend in 5 results, some more measurements were added to analyze what is happening.
A big attention was paid to determining ambient atmospheric conditions. Two calibrated
Assmann-type psychrometers were used to monitor dry and wet temperatures at both endpoints of measured distances. Fans of psychrometers were turned on immediately when stations were changed to provide satisfactory flow of air around the wet bulb. Psychrometers
were placed in EDM instrument and reflector heights, horizontally by a maximum of a couple
of meters from the instrument. Psychrometers were kept out of reach of direct sunlight. Dry
and wet temperatures were read after the 1st and after the 4th measurement of a distance (Leica
TCA2003) resp. during the 1st and during the 2nd measurement of a distance (Kern Mekometer
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ME5000). By multiple temperature determinations, readings error could be eliminated and
changes of temperatures in time could be detected. Walky-talkies were used to synchronize
readings of temperatures at both endpoints. Readings were performed with a resolution of 0.1
°C.
Air pressure was measured at the EDM instrument station only. Pressure variation along
the baseline due to the height differences or weather changes was neglected, since there is
only 4 meters maximum difference in height between the pillars. Maximal difference in pressure is therefore very small, about 0.5 hPa. In addition, this difference will be removed in
evaluation process because all distances were measured in both directions. To measure atmospheric pressure, two mechanical aneroid barometers were used to eliminate errors and
improve results. Readings of both barometers were performed after the 5th measurement of a
distance (Leica TCA2003) resp. between the 1st and the 2nd measurement of a distance (Kern
Mekometer ME5000). Before readings, pointers of barometers were arranged to zero. Readings were performed with a resolution of 0.1 hPa. The barometers were compared with the
mercury barometer of the FGI before and after the calibration of the EDM instrument. Means
of these two comparisons were used to obtain final corrections.
Partial water vapor pressure must be determined for computation of the first velocity correction. It can be evaluated from measurements of dry and wet temperatures and air pressure
which is the way how it is standardly done by the FGI. But because there was available also a
digital instrument for measuring relative humidity, borrowed from Aalto University, it was
used for comparison (Leica TCA2003 only). Measurements were done at EDM instrument
stations. Reading resolution was 0.1 %.
4.4 Evaluation process
4.4.1 Software
For computations, the combination of two mathematically orientated computer programs was
used. Microsoft Excel 2010 (version 14.0.6112.5000) and Mathworks Matlab R2011b (version 7.13.0.564) were the main software tools used to perform calculations. Microsoft Excel
was picked for its well-arranged and easily readable spreadsheet data structure. Observed data
and pre-processed data, recorded in an Excel file with a table structure, are easily readable,
understandable and accessible for everyone who has a computer with installed Microsoft Ex-
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cel software. Mathworks Matlab was chosen for its advanced and scientific functions enabling
to create a specific piece of program code or a script to perform requested operations. As a
source of data for the script, the previously created Excel file was used. Graphical results were
also generated from Matlab software. Advantageous connection of both of these software enabled that numerical results could be automatically written into the Excel file after all computations in Matlab software were finished.
4.4.2 Data pre-processing in Excel
All observed data were originally recorded in an analogue written form in notebooks. They
were retyped into Excel files. Each calibration has its own Excel file with several different
sheets (Appendix 3). Files are named according to a date or dates of measurement. The first
sheet “info” contains information about calibration (calibrated EDM instrument, used reflector, calibration baseline, date of measurement, weather conditions) and necessary constants
needed for further computations (EDM carrier wavelength, reference conditions, reference
refractive index, heights of instruments, initial instrumental corrections, psychrometer constant, radius of the Earth).
The second sheet “observation” gathers all observed data and provides initial preprocessing. The first table shows all measured distances and computes arithmetic means of
distances, differences (distance
mean) and type A uncertainties of means according to the
EA 4/02 [19] and the GUM [20]. The second table processes the temperature observations.
Arithmetic means of two observations of dry and wet temperatures at both endpoints of distances are evaluated and consequently average temperatures along the lines are computed as
arithmetic means of temperatures obtained at both endpoints. The third table records observations of air pressures, introduces corrections of used aneroid barometers and computes final
pressures as arithmetic means of data obtained from two barometers.
The third sheet “baseline” shows all baseline data (Appendix 6) which are used for further computations (pillar heights, projection corrections, true distances). These data are not
obtained from measurements with EDM instrument, but they are known from previous calibration of the baseline. These data should be as up-to-date as possible to reach the best possible results of calibration.
The fourth sheet “p.w.v.p.” computes and compares partial water vapor pressures and relative humidities. In the first table, source data are processed. When there are measurements of
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relative humidity from relative humidity meter, the second table computes partial water vapor
pressures from relative humidity, dry temperatures and air pressures. All calculations meet
equations stated in subsection 2.6.3. The third table compares both calculated partial water
vapor pressures and shows how differences could affect the distance. All the tables were used
to compare two methods of obtaining the partial water vapor pressure. Since the differences
were significant and the used relative humidity meter has no calibration certificate, measurements of relative humidity were not used in further computations. In fact, the Matlab script
computes partial water vapor pressures itself, so tables in this sheet have only informative
character.
The fifth sheet “temperature” compares differences between average temperatures at both
endpoints. Differences are computed as temperature at the EDM instrument minus temperature at the reflector station. Differences are intended to be used for analysis of uncertainties of
measurement. Unfortunately, two psychromoters were not always placed in the same way. It
means sometimes their positions were interchanged due to some inattention of people operating them. This makes no affection to the results of calibrations but it makes the analysis of
uncertainties more complicated. Small systematic error, which is present in every measurement between both psychrometers, has now sometimes an opposite mathematical sign. Since
the systematic error is very small and the interchanged positions were recorded, it makes the
analysis possible.
The sixth sheet “processed” gathers all previously computed data which are handled to
Matlab for further computations. Data in this table are mean distances, dry and wet temperatures and air pressures. The seventh sheet “results” shows numerical results computed in
Matlab. There is the additive constant, the scale correction and the root-mean-square deviation calculated separately and using simple linear regression. The root-mean-square deviation
of the cubic regression is also present. Coefficients of cubic regressions are not recorded since
they are not very interesting for practical purposes. The third table shows some partial results
of computations from Matlab as velocity corrections, geometrical corrections, measured minus true distances and residuals. The eighth sheet “uncertainties” contains a table created according to the EA 4/02 [19] describing all identified uncertainties in the measurement process.
Creation of this uncertainty budget of measurement is described in subsection 4.4.4.
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4.4.4 Data processing in Matlab
The core of the script (Appendix 7), programmed in Mathworks Matlab R2011b, is the file
“main”. It is the only runnable file, the other files are functions which are automatically called
by the main program. The function “ac” computes the additive constant, the function “sc”
computes the scale correction, the function “reg” computes simple linear regression and cubic
regression, the function “graph” plots all graphical outputs. Every piece of program code has
been written in a way to be valid for all calibrations on the Nummela Standard Baseline. The
only thing, which has to be changed before running the script, is the name of the Excel data
source file. After few corrections, the program code may be also usable to be applied for
evaluations of measurements performed on other calibration baselines with different measurement procedures.
The file “main” is divided into cells according to logical reasons. In the first cell, there
are information about purpose and release of the script. Information about version of Matlab,
in which the script was created, is also included. The second cell identifies the Excel data
source file, from which all input data will be extracted. This name should be changed according to a calibration set which is desired to be evaluated. The third cell imports the data from
the chosen Excel file into Matlab memory. Data from sheets “info”, “baseline” and “processed” are imported. The cells from fourth to ninth performed requested calculations. The
last tenth cell writes numerical results back into the Excel source file in the sheet “results”,
create graphs by calling the function “graph” and displays some information into the Matlab
active terminal window. When this appears in the Matlab, the program has performed all requested operations and has been successfully terminated.
The fourth cell deals with the first velocity correction. Partial water vapor pressures are
computed from psychrometer and barometer observations according to subsection 2.6.3. First
velocity corrections are evaluated according to Ciddor and Hill equations, described in subsection 2.6.4. If there is a reference refractive index given in the Excel source file (different
from zero), priority is given to this value and computed reference refractive index from referent atmospheric conditions remains unused. If the initial reference refractive index is not
known from Excel (zero value), computed value is used for evaluating first velocity corrections. Corrected distances are computed by applying these corrections to measured distances.
The fifth cell introduces the vertical geometrical correction. Station and target pillar
heights, EDM instrument and reflector heights above pillars are taken into account to calcu-
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late heights above the reference point. Height reference point of the Nummela Standard Baseline is placed at the underground marker of the pillar at 0 m. Corrections are evaluated according to equation (1.3). Corrected distances, which are horizontal distances in reference level,
are computed by applying these corrections to distances obtained in previous cell.
The sixth cell calculates the additive constant using the least squares adjustment. Baseline
true distances are not used. If there is an initial additive constant given in the Excel file (different from zero), the value is applied to all distances before calculations. Function “ac” is
called to evaluate the additive constant and its standard deviation according to subsection
3.4.2. The additive constant is applied to all distances obtained in the previous cell. The second computation of the additive constant, using corrected distances, is performed for control
reasons. The testing criterion 0.001 mm is used. If the second computed additive constant
exceeds the criterion, an error message occurs in the Matlab terminal and the program is
aborted.
The seventh cell finds the scale correction. If there is an initial scale correction given in
the Excel file (different from zero), the value is applied to all distances before calculations.
Projection corrections are added to all distances (obtained in the previous cell) and differences
between corrected measured distances and true baseline distances are computed. The scale
correction is evaluated using modified simple linear regression. The modification consists in
omission of the intercept term. Function “sc” is called to evaluate the scale correction and its
standard deviation according to subsection 3.4.2. Residuals are computed. The scale correction is applied to all distances (obtained in previous cell and modified by projection corrections). The second computation of the scale correction, using corrected distances, is
performed for control reasons. The testing criterion 0.001 mm is used. If the second computed
scale correction exceeds the criterion, an error message occurs in the Matlab terminal and the
program is aborted.
The eighth cell figures out the root-mean-square deviation from residuals obtained in the
previous cell. The ninth cell performs regression analysis. Function “reg” is called to evaluate
coefficients of the regress line according to subsection 3.4.1. The coefficients may stand for
the additive constant and the scale correction. Standard deviations of the coefficients and residuals are calculated. In addition, cubic regression model is employed to obtain the rootmean-square deviation of the cubic curve. As noted above, the tenth cell writes the numerical
results into the Excel file and creates graphs which are saved to folder “Graphs”. Five different graphs are plotted. They are titled “Additive constant analysis”, “Scale correction analy-
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sis”, “Residuals analysis”, “Simple linear regression” and “Cubic regression”. The first three
graphs are connected with the separate method and the last two graphs are connected with the
combined methods of computation of instrumental corrections.
4.4.5 Uncertainties of measurement
Evaluation of uncertainties of measurement is an essential and necessary part of every calibration. The analysis of uncertainties is processed according to the EA 4/02 [19] with respect to
the GUM [20]. The uncertainty budget of measurement was created for each calibration and it
is included in Excel files in the sheet “uncertainties”. The purpose of this analysis is to determine uncertainties of the additive constant and the scale correction. The purpose is not to determine the uncertainty of a measured distance. In that case, more uncertainties have to be
taken into account. This task is left for a user of an instrument.
The only standard uncertainty, contributing to the standard deviation associated with the
additive constant, is the standard deviation obtained from the least squares adjustment. All
other uncertainties (centering, leveling, measuring atmospheric conditions) are considered to
be already included in the uncertainty from the adjustment. On the other hand, to obtain
standard uncertainty associated with the scale correction, there are some additional quantities
which have to be included in the uncertainty budget of measurement.
All quantities, which were considered to take part in the combined standard uncertainty
(standard uncertainty associated with the output estimate), are listed in (Table 8).
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Table 8: Description of quantities contributing to the standard uncertainty
Quantity
Description
Xi
s
p
ac
sc
Ti
Nummela scale
Projection measurements
EDM additive constant
EDM scale correction
Temperature instruments
To
Temperature observations
pi
Pressure instruments
po
Pressure observations
ec
P.w.v.p. calculations
eo
P.w.v.p. observations
u
Combined std. uncertainty
Nearly all values of standard uncertainties were taken from the FGI and inspired by their long
experience with calibrating EDM instruments at the Nummela Standard Baseline. The uncertainty of the scale of the Nummela Standard Baseline is expressed as 0.10 mm/km. This value
is an estimate which comes from uncertainties of true baseline distances measured with the
Väisälä interferometer. An instability of underground markers since last interference measurements is considered negligible. The standard uncertainty of projection measurement (two
corrections for each distance) is set to be 0.07 mm. This value is non-distant dependent for
distances below 864 meters. Therefore a value of 0.081 mm/km is used by the FGI. This value may seem sort of controversial because of mixing fixed and proportional standard uncertainty. But there is no other simple way how to employ non-distance dependent standard
uncertainty in the uncertainty associated with the scale correction. Experience has shown that
this simplification is possible. Instability of pillars since last projection measurements is considered negligible.
Standard deviations of longitudinal effective temperatures
were analyzed from differ-
ences between point temperatures at both endpoints of distances. Previous experience was
also used. To compute sensitive coefficients for temperature, air pressure and partial water
vapor pressure, partial differentials of Ciddor and Hill equation (2.22) were derived. Mean
atmospheric conditions were used to obtain numerical values with sufficient accuracy.
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N L 273.15 N g p 11.27 e


T
1013.25 T 2
T2
(4.1)
N L 273.15 N g

p 1013.25 T
(4.2)
N L
11.27

e
T
(4.3)
The Ciddor and Hill closed formulae, adopted by the IAG in 1999, has some uncertainty itself. It is noted, that the closed equation deviates less than 0.25 ppm from the accurate formulae between
30 °C and
45 °C, at 1013.25 hPa and 100 % humidity. Under temperatures at
about a middle of the interval, it should be even less. But since the same equation should be
used in calibrations as well as in applications, the uncertainty of computation the first velocity
correction is not usually included in evaluation of uncertainty associated with the scale correction.
All used uncertainties of measurements are type B uncertainties. Because there are multiple observations of each distance, it offers to use also the type A evaluation of standard uncertainty. The first reason against it is the number of observations. Two observations in case of
Kern Mekometer ME5000 and even five observations in case of Leica TCA2003 are not sufficient numbers of repetition for reliable estimate of the type A uncertainty. Even if there are
thirty different distances for separate analysis, combining them is not the pure attitude to it.
The main reason against using the type A evaluation of standard uncertainty of a multiple
measured distance in field conditions is that atmospheric conditions are very unstable. Therefore it is not possible to differentiate the influence of changing conditions and the influence of
repeatability of measurement with specified instrument. In addition, most of the difference
between measurements with precision EDM instruments is considered to be caused by unstable atmospheric conditions rather than by repeatability of devices. The repeatability of measurement may only be tested under laboratory conditions to achieve satisfactory results.
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4.4.7 Combining results of calibrations
All analysis presented above concerns results of calibrations separately. Analyses of separate calibrations are unavoidable to be computed because they give the preliminary estimate
of all values and may be used for deeper analysis of results. But when there are multiple calibrations available for processing, it is necessary to deal with it and combine partial results into
one final output. The Excel file “Results” contains summary of results of calibrations and performs calculations combining the results. In most cases, final instrumental corrections are
computed as equally weighted average of all partial calibrations (Table 19). In special cases,
when weather conditions vary a lot and there are large differences between standard deviations from adjustments, weighted average may be used. Standard deviations from adjustment
are used as weights.
When there is only one calibration of an instrument, standard deviation from adjustment
is the only accuracy information available. When there are multiple calibrations, new accuracy information occurs. Standard deviations from adjustments are a good tool to evaluate
weights of calibrations but using them to evaluate final standard deviation is not the proper
way to do it. Standard deviations from adjustments reflect fitting of instrumental corrections
only relatively and do not give the absolute information. Two results may have huge difference between each other but still they will both have small standard deviations from adjustments. Therefore the experimental standard deviation is computed from results of partial
calibrations (Table 19). The more calibrations are available, the more reliable estimate of experimental standard deviation is computed. Only two calibrations do not always give satisfactory result, therefore the experimental standard deviations have to be corrected according to
previous experience with specified instrument (Table 20). In this case, values of Leica
TCA2003 were not corrected because they give reasonable estimates, but values of Kern Mekometer ME5000 were corrected according to experience of the FGI.
New uncertainty budgets of measurement have to be processed (Table 21, Table 22).
Principles remain the same, but standard deviations from (Table 20) are employed. Standard
deviations of observations of meteorological quantities are set to be zero because they are
considered to be already included in standard deviations of instrumental corrections. It is not
possible to separate different influences in field conditions. In addition, one value describing
accuracy of temperature observations during multiple calibrations would be a doubtful num-
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ber. Final instrumental corrections and extended (2σ) uncertainties are presented in (Table
23).
4.5 Results of calibrations
This section presents results and uncertainties of computations performed as described in previous section 4.4. The same tables may be found in relevant Excel files. Presented graphs are
saved in folder “Graphs”. Only two most interesting graphs are shown here, the other two
graphs are introduced in (Appendix 8, Appendix 9, Appendix 10, Appendix 11), one graph is
available in digital form only. Final results and comparison of results with FGI computations
are presented in the last two subsections 4.5.3 and 4.5.4. It should be noted that all graphs,
comparisons and final results in this section deal with separate computations of instrumental
corrections. Results of computations of combined methods serve only for comparisons with
separate computations. Please note, that not all decimal places of values are necessarily relevant to obtained accuracy.
4.5.1 The Leica TCA2003
Table 9: Results of computations – 2011-10-06, Leica TCA2003
Results of computations - Nummela, 2011-10-06, Leica TCA2003 (S.No. 438743)
Computation
Additive constant [mm] Scale correction [mm/km] RMS dev.
method
Value
Std. dev.
Value
Std. dev.
[mm]
Separately
0.313
0.078
2.033
0.101
0.249
Simple lin. regress.
0.346
0.079
1.977
0.168
0.248
Cubic regression
0.217
Table 10: Results of computations – 2011-10-07, Leica TCA2003
Results of computations - Nummela, 2011-10-07, Leica TCA2003 (S.No. 438743)
Computation
method
Value
Std. dev.
Value
Std. dev.
[mm]
Separately
0.231
0.091
1.347
0.115
0.284
0.304
0.089
1.222
0.190
0.281
Cubic regression
0.203
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Fig. 27: Graphs – 2011-10-06, Leica TCA2003
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Fig. 28: Graphs – 2011-10-07, Leica TCA2003
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Table 11: Uncertainty budget of measurement – 2011-10-06, Leica TCA2003
Uncertainty budget of measurement - 2011-10-06, Leica TCA2003 (S.No. 438743)
Unc. contrubution u i (y )
Std.
Sensitivity
Quantity Unit
Estimate x i
Additive
Scale
uncert.
coefficient
constant
correction
Xi
u (x i )
ci
[mm]
[mm/km]
From To
Unit
s
mm/km
0.10 1
×L
0.100
p
mm -1.627 1.627
0.07 1.157
0.081
0.313
ac
mm
0.078 1
0.078
2.033
sc
mm/km
0.101 1
×L
0.101
-1
Ti
K
0.11 0.947 × L [K ]
0.104
To
K
285.7 286.3
pi
hPa
-
po
hPa
976.9 979.2
ec
hPa
14.2
eo
hPa
15.0
14.2
15.0
0.08 0.947 × L [K-1]
0.1 0.278 × L [hPa-1]
-1
0.2 0.278 × L [hPa ]
0.1 0.039 × L [hPa-1]
-1
0.4 0.039 × L [hPa ]
u
-
0.076
-
0.028
-
0.056
-
0.003
-
0.016
0.078
0.218
Table 12: Uncertainty budget of measurement – 2011-10-07, Leica TCA2003
Uncertainty budget of measurement - 2011-10-07, Leica TCA2003 (S.No. 438743)
Std.
Sensitivity
Quantity Unit
Estimate x i
Additive
Scale
uncert.
coefficient
constant
correction
Xi
u (x i )
ci
[mm]
[mm/km]
From To
Unit
s
mm/km
0.10 1
×L
0.100
p
mm -1.627 1.627
0.07 1.157
0.081
0.231
ac
mm
0.091 1
0.091
1.347
sc
mm/km
0.115 1
×L
0.115
-1
Ti
K
0.11 0.963 × L [K ]
0.106
To
K
281.8 284.9
pi
hPa
-
po
hPa
ec
hPa
eo
hPa
974.2 976.2
8.0
8.0
9.5
9.5
0.36 0.963 × L [K-1]
0.1 0.280 × L [hPa-1]
-1
0.2 0.280 × L [hPa ]
0.0 0.040 × L [hPa-1]
-1
0.4 0.040 × L [hPa ]
u
-
0.347
-
0.028
-
0.056
-
0.002
-
0.016
0.091
0.407
Measurements with the Leica TCA2003 were held under very different weather conditions. During the first calibration on the 6th October 2011, there was cloudy and rainy whole
day. On the other hand, during the second calibration on the 7th October 2011, there was clear
and partly cloudy. Moderate wind helped to mix differently heated air layers, but still much
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bigger differences up to 1 mm were obtained in five consecutive measurements of distances.
Differences in dry temperatures at both endpoints also reached much bigger values, more than
1 °C. It causes not only minor increase of computed standard uncertainties but also significant
difference in computed instrumental correction, especially in the scale correction. It may seem
as a problem, but in fact it can be an advantage. Diverse weather conditions and diverse calibration results involve more possibilities of use of the instrument in praxis. The average of the
results may has larger associated uncertainty, but it gives a more complex and reliable estimate of the true instrumental correction.
4.5.2 The Kern Mekometer ME5000
Table 13: Results of computations – 2011-11-24 and 25, Kern ME5000
Results of computations - Nummela, 2011-11-24 and 25, Kern ME5000 (S.No. 357094)
Computation
method
Value
Std. dev.
Value
Std. dev.
[mm]
Separately
0.023
0.050
0.209
0.068
0.168
0.115
0.051
0.051
0.108
0.159
Cubic regression
0.156
Table 14: Results of computations – 2011-11-25 and 28, Kern ME5000
Results of computations - Nummela, 2011-11-25 and 28, Kern ME5000 (S.No. 357094)
Computation
method
Value
Std. dev.
Value
Std. dev.
[mm]
Separately
0.025
0.042
0.268
0.066
0.163
0.105
0.050
0.131
0.106
0.156
Cubic regression
0.156
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Fig. 29: Graphs – 2011-11-24 and 25, Kern Mekometer ME5000
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Fig. 30: Graphs – 2011-11-25 and 28, Kern Mekometer ME5000
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Table 15: Uncertainty budget of measurement – 2011-11-24 and 25, Kern ME5000
Uncertainty budget of measurement - 2011-11-24 and 25, Kern ME5000 (S.No. 357094)
Std.
Sensitivity
Quantity Unit
Estimate x i
Additive
Scale
uncert.
coefficient
constant
correction
Xi
u (x i )
ci
[mm]
[mm/km]
From To
Unit
s
mm/km
0.10 1
×L
0.100
p
mm -1.627 1.627
0.07 1.157
0.081
0.023
ac
mm
0.050 1
0.050
0.209
sc
mm/km
0.068 1
×L
0.068
-1
Ti
K
0.11 1.031 × L [K ]
0.113
To
K
278.7 280.5
pi
hPa
-
po
hPa
ec
hPa
eo
hPa
994.3 999.7
8.4
10.2
8.4
10.2
0.10 1.031 × L [K-1]
0.1 0.289 × L [hPa-1]
-1
0.2 0.289 × L [hPa ]
0.0 0.040 × L [hPa-1]
-1
0.4 0.040 × L [hPa ]
u
-
0.103
-
0.029
-
0.058
-
0.002
-
0.016
0.050
0.222
Table 16: Uncertainty budget of measurement – 2011-11-25 and 28, Kern ME5000
Uncertainty budget of measurement - 2011-11-25 and 28, Kern ME5000 (S.No. 357094)
Std.
Sensitivity
Quantity Unit
Estimate x i
Additive
Scale
uncert.
coefficient
constant
correction
Xi
u (x i )
ci
[mm]
[mm/km]
From To
Unit
s
mm/km
0.10 1
×L
0.100
p
mm -1.627 1.627
0.07 1.157
0.081
0.025
ac
mm
0.042 1
0.042
0.268
sc
mm/km
0.066 1
×L
0.066
-1
Ti
K
0.11 1.030 × L [K ]
0.113
To
K
274.6 281.3
pi
hPa
-
po
hPa
ec
hPa
eo
hPa
975.9 993.0
5.0
5.0
10.7
10.7
0.17 1.030 × L [K-1]
0.1 0.291 × L [hPa-1]
-1
0.2 0.291 × L [hPa ]
0.0 0.041 × L [hPa-1]
-1
0.4 0.041 × L [hPa ]
u
-
0.175
-
0.029
-
0.058
-
0.002
-
0.016
0.042
0.263
Two calibrations of the Kern Mekometer ME5000 were held in three days because of
more complicated operation with the instrument and short November days in Finland. During
the first two days there was quite similar weather, cloudy, rainy and drizzly. The third day it
was cloudy and then clear, dry temperatures were lower than two days before. According to
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experience of the FGI, Kern Mekometer ME5000 is less sensitive to weather changes (especially sunlight) than Leica TCA2003. One reason is its long time of single measurement
which enables to mediate small changes of atmospheric conditions. Differences in dry temperatures were up to 0.6 °C in all three days. It all causes similar standard deviations from
adjustment. The additive constant was practically the same at both calibrations, the scale correction differs slightly.
4.5.3 Summary and final results
Following table gives well-arrange view of the most important results which were already
presented in previous subsections 4.5.1 and 4.5.2.
Table 17: Summary of results of calibrations
EDM
Leica
TCA2003
Kern
ME5000
Summary of results of calibrations
Add. const. [mm]
Scale corr. [mm/km]
Calib.
Value
Std. dev.
Value
Std. dev.
st
0.313
0.078
2.033
0.101
1
nd
0.231
0.091
1.347
0.115
2
st
0.023
0.050
0.209
0.068
1
nd
0.025
0.042
0.268
0.066
2
Table 18: Summary of uncertainties according to EA 4/02
EDM
Leica
TCA2003
Kern
ME5000
Summary of uncertainties (EA 4/02)
Standard uncertainty
Calib.
Add. const. [mm] Scale corr. [mm/km]
st
0.078
0.218
1
nd
0.091
0.407
2
st
0.050
0.222
1
nd
0.042
0.263
2
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Following tables deals with final instrumental corrections and associated uncertainties of
measurement which come into existence by combining partial calibrations according to subsection 4.4.5.
Table 19: Final instrumental corrections and experimental std. deviations
Final instrumental corrections and exp. std. deviations
Add. const. [mm]
Scale corr. [mm/km]
EDM instrument
Value
Std. dev.
Value
Std. dev.
Leica TCA2003
0.272
0.041
1.690
0.343
Kern ME5000
0.024
0.001
0.238
0.030
Table 20: Final instrumental corrections and modified standard deviations
Final instrumental corrections and mod. standard deviations
Add. const. [mm]
Scale corr. [mm/km]
EDM instrument
Value
Std. dev.
Value
Std. dev.
Leica TCA2003
0.272
0.041
1.690
0.343
Kern ME5000
0.024
0.020
0.238
0.080
Table 21: Uncertainty budget of measurement – Leica TCA2003
Uncertainty budget of measurement - Leica TCA2003 (S.No. 438743)
Std.
Sensitivity
Quantity Unit
Estimate x i
Additive
Scale
uncert.
coefficient
constant
correction
Xi
u (x i )
ci
[mm]
[mm/km]
From To
Unit
s
mm/km
0.10 1
×L
0.100
p
mm -1.627 1.627
0.07 1.157
0.081
0.272
ac
mm
0.041 1
0.041
1.690
sc
mm/km
0.343 1
×L
0.343
-1
Ti
K
0.11 0.959 × L [K ]
0.106
To
K
281.8 286.3
pi
hPa
-
po
hPa
ec
hPa
eo
hPa
974.2 979.2
8.0
8.0
15.0
15.0
0.00 0.959 × L [K-1]
0.1 0.280 × L [hPa-1]
-1
0.0 0.280 × L [hPa ]
0.1 0.040 × L [hPa-1]
-1
0.0 0.040 × L [hPa ]
u
103
-
0.000
-
0.028
-
0.000
-
0.004
-
0.000
0.041
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Table 22: Uncertainty budget of measurement – Kern ME5000
Uncertainty budget of measurement - Kern ME5000 (S.No. 357094)
Std.
Sensitivity
Quantity Unit
Estimate x i
Additive
Scale
uncert.
coefficient
constant
correction
Xi
u (x i )
ci
[mm]
[mm/km]
From To
Unit
s
mm/km
0.10 1
×L
0.100
p
mm -1.627 1.476
0.07 1.157
0.081
0.024
ac
mm
0.020 1
0.020
0.238
sc
mm/km
0.080 1
×L
0.080
Ti
K
0.11 1.034 × L [K-1]
0.114
To
K
274.6 281.3
pi
hPa
-
0.00 1.034 × L [K-1]
0.1 0.291 × L [hPa-1]
-1
po
hPa
ec
hPa
5.0
10.7
0.0 0.291 × L [hPa ]
0.1 0.041 × L [hPa-1]
eo
hPa
5.0
10.7
0.0 0.041 × L [hPa-1]
975.9 999.7
u
-
0.000
-
0.029
-
0.000
-
0.004
-
0.000
0.020
0.192
Table 23: Final instrumental corrections and extended uncertainties
Final instrumental corrections and extended uncertainties
Add. const. [mm]
Scale corr. [mm/km]
EDM
instrument
Value
± (2σ)
Value
± (2σ)
Leica TCA2003
0.272
0.083
1.690
0.764
Kern ME5000
0.024
0.040
0.238
0.383
4.5.4 Comparison with FGI results
Lic. Sc. Jorma Jokela from the FGI has provided results of calibrations evaluated in the official FGI fortran-programmed script. Also partial results of calibration were available. Comparison of results is presented in (Table 24). From analysis of partial results was discovered,
that the only difference between both calculations is in the computation of the first velocity
correction. Ciddor and Hill solution was used in both cases. But whereas in this paper the first
velocity correction was computed according to closed IAG formulas (subsection 2.6.4), the
FGI computes the first velocity correction according to a piece of Australian fortran program
code which employs precise Ciddor and Hill solution. Anyways the results remain very close
to each other.
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Table 24: Comparison with FGI results
Comparison with FGI results
Additive constant [mm]
Scale correction [mm/km]
EDM
Calib.
Computed
FGI results
Computed
FGI results
Value Std. dev. Value Std. dev. Value Std. dev. Value Std. dev.
st
0.313 0.078 0.314 0.080 2.033 0.101 2.001 0.098
Leica
1
nd
TCA2003 2
0.231 0.091 0.230 0.092 1.347 0.115 1.304 0.112
Kern
ME5000
1st
0.023
0.050
0.022
0.051
0.209
0.068
0.189
0.066
2nd
0.025
0.042
0.024
0.042
0.268
0.066
0.260
0.064
The calibrated Kern Mekometer ME5000 is being used as a scale transfer instrument by
the FGI. Therefore its instrumental corrections are continuously monitored in time. Lic. Sc.
Jorma Jokela from the FGI has provided results of previous calibrations of the instrument.
Five calibrations were performed during a period of 16.5.-15.6.2011. Computed instrumental
corrections in this paper are compared with previous results of calibrations in (Table 25).
Table 25: Comparison with previous calibrations of Kern ME5000
Kern ME5000 calibrations (S.No. 357094)
Additive constant [mm]
Scale correction [mm/km]
16.5.-15.6.2011 24.-28.11.2011 16.5.-15.6.2011 24.-28.11.2011
0.039
0.024
0.181
0.238
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5. Nummela and Koštice baselines
5. NUMMELA AND KOŠTICE BASELINES
In this chapter, it is intended to compare two calibration baselines and calibration procedures.
The Nummela Standard Baseline in Finland is confronted with the Czech state long distances
measuring standard Koštice. Even if the Czech state long distances measuring standard
Koštice is the official name of the baseline, further in the text it will be called the Koštice
Baseline for short. Provided data here are more or less informative and they are not official
since a part of them comes from consultations with Lic. Sc. Jorma Jokela and Ing. Jiří Lechner, CSc. In case of the Nummela Standard Baseline the data are up to date, but in case of the
Koštice baseline they were mostly collected during processing [2] in 2009/2010.
5.1 The Koštice Baseline
The baseline was constructed between years 1979 and 1980 near a village Koštice in Louny
region (Fig. 31). In 2006 VÚGTK was determined to provide research and development at the
baseline with the view of declaration of a new Czech state measuring standard. Interlaboratory comparison was carried out by the Universität der Bundeswehr München. On 26th
February 2008 the Koštice Baseline was announced as the new Czech state long distances
etalon.
Fig. 31: Schematic location and design of the Koštice Baseline
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The Koštice Baseline consists of 12 concrete-steel pillars with forced centering which are
placed in a line along a road Koštice – Libčeves (Fig. 31). Stabilizations of pillars reach into
depth from 5 to 9 meters. 66 distances between 25 and 1450 meters are realized by the pillars
(Fig. 32). A set of fixing screws complements the baseline (Fig. 33). A vertical axis of a
unique fixing screw, placed in a cylinder hole of forced centering, creates a geodetic point in
place of intersection with metal plate of a pillar (Fig. 34). Standard geodetic tribachs may be
mounted on the fixing screws to hold EDM instruments and reflectors.
Fig. 32: Instruments on observation pillars at the Koštice Baseline
Fig. 33: The set of fixing screws (left) and the detail of a fixing screw (right)
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Fig. 34: The section of forced centering of the Koštice Baseline
Several methods were used to determine true baseline distances, including measuring
with invar tapes and observing parallactic angles. Distances were measured in all combinations with Leica TCA2003 which is compared with etalon in München and with laser interferometer Hewlett Packard 5519A owned by VÚGTK. Least-square adjustment with
observations weighted according to an accuracy of Leica TCA2003 was used to compute true
distances. Inter-laboratory comparisons were made by Universität der Bundeswehr München
using Leica TDA5005 and Kern Mekometer ME5000. Combined standard uncertainty of the
Koštice baseline is determined to be as equation (5.1).
u  Q 0,5 mm ;1,5 mm /1000 m
where
u
… combined standard uncertainty
Q
… quadratic sum of all partial uncertainties
(5.1)
Section 5.1 uses bibliography: [2], [35], [36], [37].
5.2 Comparison of baselines, measurements and evaluations
Both baselines serve as geodetic baselines for calibrating electronic distance meters. Even if
the purpose of them is the same, their location, design, structure and accuracy is completely
different. Considering the Nummela Standard Baseline as the most accurate baseline in the
world and the Koštice Baseline as a new etalon established recently in 2008, it should be realized that the comparison of those two baselines is not a criticism of the Koštice Baseline. It is
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just a simple confrontation with a point of view of a student who had a chance to perform
calibrations at both baselines. Tables seem to be the most suitable way of comparing the baselines, measurement procedures and evaluation processes since all specifications have been
already described in this paper and in [2].
Table 26: Comparison of the Koštice Baseline and the Nummela Standard Baseline
Comparison of Koštice Baseline and Nummela Standard Baseline
Comparing
The Koštice Baseline
The Nummela Standard Baseline
Location
Czech Republic, Koštice
Finland, Nummela
Administrator
VÚGTK
The FGI
Responsible person
Ing. Jiří Lechner, CSc.
Established
2008 (1979)
1947 (1933)
Placing
Along a road, open area
Forest
Subsurface
Solid rock
Sandy ridge with glacial origin
Orientation
North-West
North-East
Sunshine protection
Temporal - umbrellas
Pernament - shelters
Weather sensors
NO
NO
Points
12
6
Design
For EDM instruments
For invar wires, Väisälä comparator
Stabilization
Concrete-steel pillars
Undergr. markers + concrete pillars
Forced centering
Fixing screws
Fixed Kern plates
Distances
66
30
Min. distance
25
24
Max. distance
1450
864
Calibration
Leica TCA2003
Väisälä interference comparator
Traceable scale
Through laser interferometer
Through quartz gauge system
Accuracy (1-σ)
0.5 mm + 1.5 mm/km
0.1 mm/km
Projection measur.
0.07 mm
Stability
Previous stability problems Max. difference 0.6 mm in 60 years
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Table 27: Comparison of measurement procedures at Koštice and Nummela baselines
Comparison of measurement procedures at Koštice and Nummela baselines
Comparing
The Nummela Std. Baseline
30
30
Measured distances
From first 3 pillars forward
All combinations
Horizontal distances
Slope distances
Observations of a distance
3
5
Time of one calibration
Approx. 2 hours
5-6 hours
Number of calibrations
1
Min. 2, usually 3 or more
Reflectors
More than one allowed
Always one
Sunshine protection
Uncommon
Always
Weather report
NO
YES
Tepmerature instruments
Unknown
Calibrated psychrometers
At EDM instrument only
At both endpoints
Temperature measurement
Twice per calibration
Twice per distance
Pressure instruments
Unknown
Calibrated aneroids
Pressure measurements
Twice per calibration
Once per distance
Rel. humidity instruments
None
Calibrated psychrometers
Nowhere
At both endpoints
Rel. humidity measurement
Never
Twice per distance
Table 28: Comparison of evaluation processes at Koštice and Nummela baselines
Comparison of evaluation processes at Koštice and Nummela baselines
Comparing
The Nummela Standard Baseline
First velocity correction
Instrument-determined
Ciddor and Hill formulas
Reference refractive index
Instrument-considered
From manufacturer or computed
Geometrical correction
Unknown
YES
Computation of corrections
Combined
Separately
Additive constant
Simple linear regression
Least squares adjustement
Scale correction
Simple linear regression
Modified simple lin. regression
Final results
Result of single calibration Mean of multiple calibrations
Type A uncertainty
Included
Not included
Experience-modified uncert.
Possible
Possible
Final uncertainties
Extended 2-σ (95%)
Extended 2-σ (95%)
Both baselines serve as the most accurate calibration standards in their countries and fulfill functions of national etalons. It is not fully objective to compare measuring procedures
and evaluation processes with each other because of the different accuracy of the baselines.
Less difficult and less time-consuming methods may be used in case of the less accurate baseline to achieve satisfactory results which are in congruence with baseline accuracy. Both baselines are continuously progressing for purpose of delivering the most reliable results. At the
Nummela Standard Baseline, development in the field of electronic measurement of atmos-
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pheric conditions is taking a part. The new Väisälä automatic weather station AWS330 has
been bought by the FGI to monitor conditions at the EDM instrument station. More weather
stations will be probably bought after testing period. At the Koštice Baseline, measurements
with newly purchased Leica AT401 absolute tracker are in preparations towards reducing
total uncertainties of true distances.
5.3 Research and development
This section focuses on the research and development in the field of electronic distance measurement presented by EURAMET in a document called “Towards New Absolute Long Distance Measurement Systems In Air” [38]. Nine laboratories from eight different countries,
including the FGI, MIKES and CMI, are participating in a project called “Absolute long distance measurement in air”. The new development already concerns the Nummela Standard
Baseline and it will probably concern the Koštice Baseline as well in the future. Not all scientific terms presented below are fully explained since it is not possible to give a complex description of the new techniques in such a space-saving way.
The project has the ambition to improve the current state-of-the-art in the long range distance measurements reaching a relative accuracy of 10-7. That accuracy is a challenge at present time. The FGI is the only metrological institute in the world which is currently
maintaining and developing the technique of optical interferometry with the similar accuracy.
But the realisation of such measurement is very time-consuming and requires very stable atmospheric parameters. In this project, EURAMET is trying to develop new generation of instruments making the measurement easier and basically with a higher resolution.
Challenge in the field of long distance measurement is precise knowledge of effective refractive index over a large range. To measure a local refractive index, improved absolute refractometer will be developed. The idea is to employ helium as a standard of the refractive
index in a gas refractometer to be able to monitor distortion of the cavity at different pressures
and decrease final uncertainty. Spectroscopic methods of measuring effective humidity and
temperature along the beam pass will be explored. The same beam path can be used in both
spectroscopy and length measurement allowing perfect spatial and temporal overlap of all
observations. Even though the uncertainty of the temperature measurement will be modest
compared to traditional locally measuring thermometers, even 1°C accuracy for average tem-
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perature in the beam path would be significant improvement to the existing methods in the
outdoor environment.
For well-known reasons, the classical optical interferometry is not appropriate for long
range absolute distance measurements. Ambiguity about the integer part and uncontrolled
environment causing air index fluctuations and mechanical vibrations are some of those. In
response, researchers have developed multi-wavelength interferometric measurement based
on a synthetic wavelength (spatial frequency is obtained from an interferometer injected by
two different wavelengths). Using the synthetic wavelength, which is much longer than the
optical wavelength, the integer part ambiguity and the sensitivity to air index fluctuations and
vibrations are greatly reduced. By using different synthetic wavelength, the ambiguity can be
removed and absolute distance measurement can be performed.
Within this work-package, a femtosecond frequency comb laser will be applied as a tool
for distance measurements. Two different techniques will be implemented. The first is using
the phase locking property between pulses emitted by the ultrafast laser. This allows for interference between different pulses. Frequency comb enables direct traceability from the practical realization of the meter to the definition of the second. Current frequency standards such
as atomic clocks operate in the microwave region of the spectrum, and the frequency comb
brings the accuracy of such clocks into the optical part of the electromagnetic spectrum. For
these measurements, the repetition rate and the carrier-envelope offset frequency of the comb
laser are phase locked to a reference value provided by a time standard. The main task is to
extend the measurement range from centimeters up to 100 meter and to identify the limits of
this technique. Secondly, a femtosecond laser is applied as an advanced frequency modulator
which allows using higher modulation frequencies. Methods of deriving the refractive index
of air from measurements performed with the femtosecond laser will be also explored.
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Conclusions
CONLUSIONS
The aim of the paper was to explore and describe problematic of field calibrations of electronic distance meters. Provided information can be useful for academic purposes as well as for
users of instruments who would like to meet this topic. Since distance is a fundamental physical unit and electronic distance meters are the most often used instruments for measuring distances in geodesy, learning about accuracy of these instruments should be one of the most
important matters to care about.
The author of the paper has got an opportunity to calibrate instruments at the Nummela
Standard Baseline in Finland which is considered to be the most accurate field calibration
standard worldwide. Thanks belong to the Finnish Geodetic Institute and especially to Lic. Sc.
Jorma Jokela. The chance to measure with Kern Mekometer ME5000, the unique instrument
widely known for its exceptional accuracy, is also very valuable experience. In addition, performed calibrations were not meaningless even if official calibration certificates were not executed. The Leica TCA2003 was calibrated for the first time in many years and computed
corrections may be used by Aalto University for academic purposes. The Kern Mekometer
ME5000, the important scale transfer instrument of the Finnish Geodetic Institute, is periodically calibrated to monitor the drift of instrumental corrections. Performed calibration was
noted in the time line of observations with this instrument at the Nummela Standard Baseline.
Because of previous calibrations on the Koštice Baseline in Czech Republic realized by
the author, there was an opportunity to compare baselines, equipment, measuring procedures
and evaluation processes, even if the same EDM instruments were not used in both cases.
Considering the long history, stability and accuracy of the Nummela Standard Baseline, the
reputation of the Finnish Geodetic Institute and deep experiences of Lic. Sc. Jorma Jokela,
pieces of knowledge from Finland could inspire the Research Institute of Geodesy, Topography and Cartography in the Czech Republic. The valuable contact with the Finnish Geodetic
Institute will enable further co-operation, for example the participation during interference
measurements at the Nummela Standard Baseline in 2013.
Current state-of-the-art technology of calibration of electronic distance meters will face
important changes in near future as the new development in this field will continue to progress. Absolute distance measurements by synthetic wavelength interferometry and by femto-
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Conclusions
second technology are developing as well as spectroscopic methods for measuring temperature and humidity along the signal path. Research is being developed by a group of European
organizations dealing with geodesy and metrology. Both Finland (FGI) and the Czech Republic (CMI) are participating in the research. Comparison of new methods with classical ones
will be held in order to verify the usability and reliability of new technologies. The Nummela
Standard Baseline will be used for these purposes.
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[38] EURAMET. Towards Nes Absolute Long Distance Measurement Systems In
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List of used abbreviations
LIST OF USED ABBREVIATIONS
AG
BIPM
CMI
CTU
DORIS
EA
EAL
EDM
EDM
EURAMET
FGI
GNSS
GUM
IEC
IFCC
ILAC
ISO
IUGG
IUPAC
IUPAP
JCGM
LSA
MIKES
OIML
RMSD
SCG
SI
SLR
VLBI
VÚGTK
WECC
Absolute Gravimetry
Bureau International des Poids et Mesures
Czech Metrology Institute
Czech Technical University in Prague
Doppler Orbitography and Radiopositioning Integrated by Satellite
European co-operation for Accreditation
Evaluation Assurance Level
Electronic Distance Measurement
Electronic Distance Meter
The European Association of National Metrology Institutes
Finish Geodetic Institute
Global Navigation Satellite System
Guide to the expression of uncertainty in measurement
International Electrotechnical Commission
International Federation of Clinical Chemistry and Laboratory Medicine
International Laboratory Accreditation Cooperation
International Organization for Standardization
International Union of Geodesy and Geophysics
International Union of Pure and Applied Chemistry
International Union of Pure and Applied Physics
Joint Committee for Guides in Metrology
Least squares adjustment
Centre for metrology and accreditation (in Finland)
International Organization of Legal Metrology
Root-mean-square deviation
Superconductivity Gravimetry
Système international d'unités
Satellite Laser Ranging
Very Long Baseline Interferometry
Research Institute of Geodesy, Topography and Cartography
Western European Calibration Cooperation
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List of figures
LIST OF FIGURES
Fig. 1: Accuracy vs. precision - graph .................................................................................... 12
Fig. 2: Accuracy vs. precision - targets .................................................................................. 13
Fig. 3: The prototype meter bar of the USA ........................................................................... 14
Fig. 4: A measuring metal chain ............................................................................................. 17
Fig. 5: Geodimeter NASM-2A ............................................................................................... 18
Fig. 6: Fizeau’s experiment .................................................................................................... 20
Fig. 7: A diagram of the electromagnetic spectrum ............................................................... 21
Fig. 8: A model of the daily cycle of the coefficient of refraction ......................................... 35
Fig. 9: The underground marker at 24 m ................................................................................ 65
Fig. 10: The observation pillars at 0 m (left) and 864 m (right) ............................................... 66
Fig. 11: The Kern-type forced-centering system ...................................................................... 66
Fig. 12: Quartz gauges (left) and quartz gauge comparator (right) at Tuorla Observatory ...... 68
Fig. 13: Comparisons of the length of quartz gauge no. VIII at Tuorla Observatory .............. 69
Fig. 14: The principle of the Väisälä interference comparator (interference 0-6-24) .............. 70
Fig. 15: Instruments of the Väisälä interference comparator at the pillars 0 and 1 .................. 70
Fig. 16: The mirror equipment of the Väisälä interference comparator ................................... 71
Fig. 17: A plumbing rod above an underground marker (left), a set of adaptor bars (right) .... 73
Fig. 18: Leica TCA2003 + Wild Leitz GPH1P ........................................................................ 75
Fig. 19: Kern Mekometer ME5000 + Kern RMO5035 ............................................................ 77
Fig. 20: The reflector “Wild Leitz GPH1P” ............................................................................. 78
Fig. 21: The reflector “Kern RMO5035” ................................................................................. 79
Fig. 22: Assmann-type psychrometer “Thermometer Thies Clima” ........................................ 79
Fig. 23: The aneroid barometers "Thomen altimeters" ............................................................ 80
Fig. 24: The relative humidity meter “Vaisala HM 34” ........................................................... 81
Fig. 25: Calibrating the Leica TCA2003 + the Wild Leitz GPH1P ......................................... 83
Fig. 26: Calibrating the Kern Mekometer ME5000 + the Kern RMO5035 ............................. 83
Fig. 27: Graphs – 2011-10-06, Leica TCA2003 ....................................................................... 95
Fig. 28: Graphs – 2011-10-07, Leica TCA2003 ....................................................................... 96
Fig. 29: Graphs – 2011-11-24 and 25, Kern Mekometer ME5000 .......................................... 99
Fig. 30: Graphs – 2011-11-25 and 28, Kern Mekometer ME5000 ........................................ 100
Fig. 31: Schematic location and design of the Koštice Baseline ............................................ 106
Fig. 32: Instruments on observation pillars at the Koštice Baseline ...................................... 107
Fig. 33: The set of fixing screws (left) and the detail of a fixing screw (right)...................... 107
Fig. 34: The section of forced centering of the Koštice Baseline .......................................... 108
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List of tabels
LIST OF TABELS
Table 1: Distance errors caused by omission of humidity ..................................................... 31
Table 2: The Nummela baseline lengths during the years 1947-2007 ................................... 72
Table 3: Constants used for calculations – Leica TCA2003 .................................................. 75
Table 4: Constants used for calculations – Kern Mekometer ME5000 .................................. 77
Table 5: Overview of measurements ...................................................................................... 82
Table 6: Ranges of atmospheric conditions during calibrations ............................................ 82
Table 7: Order of measured distances .................................................................................... 84
Table 8: Description of quantities contributing to the standard uncertainty .......................... 91
Table 9: Results of computations – 2011-10-06, Leica TCA2003 ......................................... 94
Table 10: Results of computations – 2011-10-07, Leica TCA2003 ......................................... 94
Table 11: Uncertainty budget of measurement – 2011-10-06, Leica TCA2003 ...................... 97
Table 12: Uncertainty budget of measurement – 2011-10-07, Leica TCA2003 ...................... 97
Table 13: Results of computations – 2011-11-24 and 25, Kern ME5000 ................................ 98
Table 14: Results of computations – 2011-11-25 and 28, Kern ME5000 ................................ 98
Table 15: Uncertainty budget of measurement – 2011-11-24 and 25, Kern ME5000 ........... 101
Table 16: Uncertainty budget of measurement – 2011-11-25 and 28, Kern ME5000 ........... 101
Table 17: Summary of results of calibrations ......................................................................... 102
Table 18: Summary of uncertainties according to EA 4/02 ................................................... 102
Table 19: Final instrumental corrections and experimental std. deviations ........................... 103
Table 20: Final instrumental corrections and modified standard deviations .......................... 103
Table 21: Uncertainty budget of measurement – Leica TCA2003 ......................................... 103
Table 22: Uncertainty budget of measurement – Kern ME5000 ........................................... 104
Table 23: Final instrumental corrections and extended uncertainties .................................... 104
Table 24: Comparison with FGI results ................................................................................. 105
Table 25: Comparison with previous calibrations of Kern ME5000...................................... 105
Table 26: Comparison of the Koštice Baseline and the Nummela Standard Baseline ........... 109
Table 27: Comparison of measurement procedures at Koštice and Nummela baselines ....... 110
Table 28: Comparison of evaluation processes at Koštice and Nummela baselines .............. 110
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List of appendixes
LIST OF APPENDIXES
Appendix 1: An example of field records of observations – Leica TCA2003 ....................... 123
Appendix 2: An example of field records of observations – Kern Mekometer ME5000 ...... 124
Appendix 3: Examples of processing observations – 2011-10-06, Leica TCA2003 ............. 125
Appendix 4: Processed observations – Leica TCA2003 ........................................................ 127
Appendix 5: Processed observations – Kern Mekometer ME5000 ........................................ 129
Appendix 6: Data of the Nummela Standard Baseline ........................................................... 131
Appendix 7: The computation scripts for Matlab R2011b (7.13.0.564) ................................ 132
Appendix 8: Graphs – 2011-10-06, Leica TCA2003 ............................................................. 141
Appendix 9: Graphs – 2011-10-07, Leica TCA2003 ............................................................. 142
Appendix 10: Graphs – 2011-11-24 and 25, Kern Mekometer ME5000 ............................... 143
Appendix 11: Graphs – 2011-11-25 and 28, Kern Mekometer ME5000 ............................... 144
Appendix 12: Example calibration certificate issued by the FGI ........................................... 145
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List of attachments
LIST OF ELECTRONIC ATTACHMENTS
All electronic attachments are stored on the enclosed CD-ROM.
Attachment 1: Computations
Attachment 2: Materials
Attachment 3: Figures
Attachment 4: Photos
Attachment 5: Others
– Numerical computations and graphical outputs
– Used and related materials
– Used figures
– Photographs from measurements
– Other unspecified files
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Appendixes
Appendix 1: An example of field records of observations – Leica TCA2003
a) At the EDM instrument station
b) At the reflector station
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Appendixes
Appendix 2: An example of field records of observations – Kern Mekometer ME5000
a) At the EDM instrument station
b) At the reflector station
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Appendixes
Appendix 3: Examples of processing observations – 2011-10-06, Leica TCA2003
Distance observations - Nummela, 2011-10-06, Leica TCA2003 (S.No. 438743)
Pillar
Slope distance Mean distance Difference Uncertainty type A
Station Target
[m]
[m]
[mm]
[mm]
24.0349
-0.06
24.0350
0.04
0
24
24.0350
24.03496
0.04
0.02
24.0349
-0.06
24.0350
0.04
Temperature observations - Nummela, 2011-10-06, Leica TCA2003 (S.No. 438743)
Pillar
Dry (S) [°C] Dry (T) [°C] Means (S) [°C] Mean D (S-T) [°C]
Station (S) Target (T) Wet (S) [°C] Wet (T) [°C] Means (T) [°C] Mean W (S-T) [°C]
12.5
12.6
12.45
12.53
12.2
12.4
12.20
0
24
--------------- --------------- ----------------------------12.4
12.6
12.60
12.30
12.2
12.4
12.40
Pressure observations - Nummela, 2011-10-06, Leica TCA2003 (S.No. 438743)
Pillar
Measured Correction Corrected Difference Mean
Station Target
[hPa]
[hPa]
[hPa]
[hPa]
[hPa]
988.9
0
24
-9.9
979.0
--------------- --------------- --------------991.4
-12.1
-0.3
979.15
979.3
Partial water vapour pressure from psychrometer - 2011-10-06, Leica TCA2003
Pillar
Dry temp. Wet. temp. Pressure Sat. w.v.p. Part. w.v.p. Rel. hum.
Station Target
t [°C]
t´ [°C]
p [hPa] E´ [hPa]
e [hPa]
h [%]
24
12.53
12.30
979.15
14.4
14.2
98
72
12.65
12.38
979.10
14.4
14.3
97
0
216
12.60
12.45
979.05
14.5
14.4
98
432
12.75
12.50
979.05
14.5
14.4
97
864
12.68
12.45
979.00
14.5
14.4
98
125
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Appendixes
Partial water vapour pressure from rel. hum. - 2011-10-06, Leica TCA2003
Pillar
Dry temp. Pressure Rel. hum. Sat. w.v.p. Part. w.v.p.
Station Target
t [°C]
p [hPa]
h [%]
E [hPa]
e [hPa]
24
12.53
979.15
84.6
14.6
12.3
72
12.65
979.10
86.6
14.7
12.7
0
216
12.60
979.05
87.8
14.6
12.9
432
12.75
979.05
88.9
14.8
13.1
864
12.68
979.00
89.6
14.7
13.2
Differences of partial w.v.p. - 2011-10-06, Leica TCA2003
Pillar
Partial w.v.p. [hPa] Difference Distance error
Station Target
psych. rel. hum.
[hPa]
[ppm]
24
14.2
12.3
1.9
0.08
72
14.3
12.7
1.5
0.06
0
216
14.4
12.9
1.5
0.06
432
14.4
13.1
1.2
0.05
864
14.4
13.2
1.2
0.05
Comparison of dry temperatures - 2011-10-06
Pillar
Dry temperatures [°C]
Station (S) Target (T) Station (S) Target (T) Diff (S-T)
24
12.45
12.60
-0.15
72
12.70
12.60
0.10
0
216
12.60
12.60
0.00
432
12.75
12.75
0.00
864
12.75
12.60
0.15
126
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Appendixes
Appendix 4: Processed observations – Leica TCA2003
a) 2011-10-06 (1st calibration)
Processed observations - Nummela, 2011-10-06, Leica TCA2003
Pillar
Distance Dry temp. Wet temp. Pressure
Station Target
d´ [m]
t [°C]
t´ [°C]
p [hPa]
24
24.03496 12.53
12.30
979.15
72
72.01490 12.65
12.38
979.10
0
216
216.05456 12.60
12.45
979.05
432
432.09458 12.75
12.50
979.05
864
864.12268 12.68
12.45
979.00
0
24.03496 13.05
12.83
978.50
72
47.97968 12.90
12.73
978.55
24
216
192.01920 12.85
12.70
978.45
432
408.05930 12.83
12.68
978.70
864
840.08764 12.68
12.53
978.75
0
72.01452 13.00
12.80
977.95
24
47.97944 13.08
12.88
978.00
72
216
144.03950 13.00
12.85
978.00
432
360.07944 13.05
12.90
978.00
864
792.10714 13.00
12.88
977.80
0
216.05418 13.10
12.93
977.70
24
192.01914 13.10
13.00
977.70
216
72
144.03950 13.10
12.95
977.75
432
216.03950 13.08
12.93
977.85
864
648.06716 13.00
12.80
977.75
0
432.09420 13.18
13.00
977.70
24
408.05924 13.15
12.98
977.75
432
72
360.07936 13.18
12.95
977.70
216
216.03944 13.05
12.93
977.50
864
432.02696 13.05
12.95
977.25
0
864.12118 13.05
12.88
976.85
24
840.08622 13.00
12.85
977.05
864
72
792.10636 13.05
12.88
977.20
216
648.06670 13.10
12.85
977.15
432
432.02706 13.08
12.95
977.15
127
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Appendixes
b) 2011-10-07 (2nd calibration)
Processed observations - Nummela, 2011-10-07, Leica TCA2003
Pillar
Station Target
d´ [m]
t [°C]
t´ [°C]
p [hPa]
24
24.03548
8.88
7.08
974.20
72
72.01516
8.98
7.18
974.15
0
216
216.05530
8.85
7.28
974.15
432
432.09590
8.63
7.38
974.25
864
864.12520
9.18
7.40
974.35
0
24.03512 10.08
7.78
974.70
72
47.97980
9.95
7.60
974.75
24
216
192.02000
9.58
7.60
974.70
432
408.06068
9.33
7.55
974.55
864
840.09024
9.25
7.43
974.55
0
72.01482 10.00
7.45
975.05
24
47.97972 10.20
7.60
975.00
72
216
144.04010
9.93
7.60
975.00
432
360.08048 10.13
7.50
975.00
864
792.10948 10.23
7.50
975.15
0
216.05478
9.98
7.48
975.65
24
192.01978 10.13
7.48
975.60
216
72
144.03998 10.08
7.43
975.55
432
216.04014 10.13
7.60
975.40
864
648.06940 10.18
7.53
975.45
0
432.09504 11.28
7.65
976.00
24
408.06006 11.38
7.58
976.10
432
72
360.08012 11.30
7.88
976.00
216
216.03982 11.25
8.00
976.00
864
432.02782 11.73
8.35
975.95
0
864.12326 11.70
7.78
976.15
24
840.08822 11.53
7.68
976.15
864
72
792.10822 11.75
7.78
976.15
216
648.06794 11.53
8.00
976.15
432
432.02800 11.60
8.05
976.20
128
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Appendixes
Appendix 5: Processed observations – Kern Mekometer ME5000
a) 2011-11-24 and 25 (1st calibration)
Processed observations - 2011-11-24 and 25, Kern ME5000
Pillar
Station Target
d´ [m]
t [°C]
t´ [°C]
p [hPa]
24
24.03530
7.25
5.88
999.70
72
72.01605
6.73
5.95
999.30
0
216
216.05860
6.55
6.08
998.95
432
432.10200
6.43
6.13
998.75
864
864.13675
6.50
6.23
998.65
0
24.03530
6.73
6.50
997.35
72
47.98035
6.75
6.50
997.85
24
216
192.02290
6.65
6.43
998.00
432
408.06615
6.68
6.45
998.05
864
840.10105
6.55
6.30
998.20
0
72.01560
6.98
6.73
996.90
24
47.98050
7.00
6.75
997.00
72
216
144.04255
6.95
6.73
997.00
432
360.08565
7.00
6.75
997.25
864
792.12035
6.83
6.58
997.50
0
216.05815
6.25
5.93
998.05
24
192.02300
6.20
5.90
998.00
216
72
144.04235
6.08
5.85
997.80
432
216.04340
6.15
5.98
997.65
864
648.07825
6.40
6.13
997.60
0
432.10195
5.68
5.53
996.35
24
408.06670
5.58
5.45
995.85
432
72
360.08600
5.70
5.58
995.75
216
216.04340
5.75
5.68
995.55
864
432.03460
6.00
5.95
995.20
0
864.13475
7.35
7.30
994.30
24
840.09990
6.95
6.88
994.45
864
72
792.11965
6.78
6.73
994.65
216
648.07740
6.58
6.48
994.50
432
432.03465
6.33
6.28
995.15
129
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Appendixes
b) 2011-11-25 and 28 (2nd calibration)
Processed observations - 2011-11-25 and 28, Kern ME5000
Pillar
Station Target
d´ [m]
t [°C]
t´ [°C]
p [hPa]
24
24.03525
7.73
7.58
993.00
72
72.01570
7.80
7.68
992.75
0
216
216.05785
7.93
7.80
992.60
432
432.10040
7.93
7.78
992.40
864
864.13385
8.00
7.85
992.30
0
24.03520
8.13
7.98
990.40
72
47.98015
8.18
7.95
990.20
24
216
192.02230
8.00
7.80
990.20
432
408.06475
8.00
7.75
990.15
864
840.09790
7.95
7.63
990.05
0
72.01540
7.63
6.95
989.25
24
47.98040
7.68
7.08
989.50
72
216
144.04210
7.65
7.25
989.80
432
360.08460
7.70
7.38
989.90
864
792.11770
7.88
7.55
990.05
0
216.05790
1.55
0.38
975.85
24
192.02270
1.93
0.40
975.85
216
72
144.04215
2.00
0.43
975.95
432
216.04315
1.88
0.28
976.45
864
648.07755
1.68
0.18
976.85
0
432.10150
1.75
0.10
978.85
24
408.06630
1.70
0.00
978.30
432
72
360.08560
1.68
0.00
978.10
216
216.04315
1.83
0.13
977.90
864
432.03440
1.48
0.13
977.60
0
864.13625
1.65
0.13
979.85
24
840.10110
1.63
0.18
980.10
864
72
792.12045
1.68
0.15
980.20
216
648.07805
1.78
0.30
980.35
432
432.03480
1.83
0.50
980.90
130
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Appendixes
Appendix 6: Data of the Nummela Standard Baseline
Data of the Nummela Standard Baseline - valid for 2011-10
Pillar
Pillar height [m]
Proj. corr. [mm] True distance
Station Target
Station Target
Back
Forward
D [m]
24
1.637
1.523
-1.540
-0.461
24.033218
72
1.637
1.293
-1.540
1.627
72.014950
0
216
1.637
0.600
-1.540
-0.499 216.053128
432
1.637
-0.438
-1.540
1.476 432.095283
864
1.637
-2.486
-1.540
0.827 864.122864
0
1.523
1.637
-1.540
-0.461
24.033218
72
1.523
1.293
0.461
1.627
47.981732
24
216
1.523
0.600
0.461
-0.499 192.019910
432
1.523
-0.438
0.461
1.476 408.062065
864
1.523
-2.486
0.461
0.827 840.089646
0
1.293
1.637
-1.540
1.627
72.014950
24
1.293
1.523
0.461
1.627
47.981732
72
216
1.293
0.600
-1.627
-0.499 144.038178
432
1.293
-0.438
-1.627
1.476 360.080333
864
1.293
-2.486
-1.627
0.827 792.107914
0
0.600
1.637
-1.540
-0.499 216.053128
24
0.600
1.523
0.461
-0.499 192.019910
216
72
0.600
1.293
-1.627
-0.499 144.038178
432
0.600
-0.438
0.499
1.476 216.042155
864
0.600
-2.486
0.499
0.827 648.069736
0
-0.438
1.637
-1.540
1.476 432.095283
24
-0.438
1.523
0.461
1.476 408.062065
432
72
-0.438
1.293
-1.627
1.476 360.080333
216
-0.438
0.600
0.499
1.476 216.042155
864
-0.438
-2.486
-1.476
0.827 432.027581
0
-2.486
1.637
-1.540
0.827 864.122864
24
-2.486
1.523
0.461
0.827 840.089646
864
72
-2.486
1.293
-1.627
0.827 792.107914
216
-2.486
0.600
0.499
0.827 648.069736
432
-2.486
-0.438
-1.476
0.827 432.027581
131
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Appendixes
Appendix 7: The computation scripts for Matlab R2011b (7.13.0.564)
%% 1) Software information
clc, clear all, format long g;
title='Calibration of the EDM instrument';
release='2012-01-05';
soft='Matlab 7.13.0.564 (R2011b)';
% script purpose
% YEAR-MONTH-DATE of release
% software to run the script
%% 2) Data source file identification (Microsoft Excel file)
file='111125_28.xlsx';
% Data source file
%% 3) Load input data from Microsoft Excel file (*.xls, *.xlsx file)
% Load information
[no,info] = xlsread(file,'info','B3:B7');
edm
= info(1);
%
prism
= info(2);
%
bsline = info(3);
%
date
= info(4);
%
weather = info(5);
%
EDM instrument / Total station
Reflecting prism
Calibration baseline
YEAR-MONTH-DATE of measurement
Weather conditions
% Load constants
const = xlsread(file,'info','E4:E14');
lam = const(1);
t_r = const(2);
p_r = const(3);
h_r = const(4);
n_r = const(5);
ac0 = const(6);
sc0 = const(7);
hi = const(8);
hp = const(9);
A
= const(10);
R
= const(11);
EDM carrier wavelength
[nm]
Reference temperature
[°C]
Reference atm. pressure
[hPa]
Reference rel. humidity
[%]
Reference refractive index
Initial additive constant [mm]
Initial scale correction [ppm]
EDM instrument height
[m]
Reflecting prism height
[m]
Assman psychrometer constant
Radius of the Earth
[m]
%
%
%
%
%
%
%
%
%
%
%
% Load standard baseline data
base = xlsread(file,'baseline','C5:G34');
h_s = base(:,1);
%
h_t = base(:,2);
%
c_b = base(:,3);
%
c_f = base(:,4);
%
D
= base(:,5);
%
Station pillar heights
Target pillar heights
Proj. corrections back
Proj. corrections forward
True distances
[m]
[m]
[mm]
[mm]
[m]
% Load observations
obs = xlsread(file,'processed','C5:F34');
d
= obs(:,1);
%
t_d = obs(:,2);
%
t_w = obs(:,3);
%
p
= obs(:,4);
%
Mean
Mean
Mean
Mean
[m]
[°C]
[°C]
[hPa]
of
of
of
of
all distances
dry temperatures
wet temperatures
atm. pressures
%% 4) The first velocity correction
% Saturated water vapor pressures [hPa]
Ew = (1.0007+3.46e-6.*p).*6.1121.*exp(17.502.*t_w./(240.97+t_w)); % For t_w
Ed = (1.0007+3.46e-6.*p).*6.1121.*exp(17.502.*t_d./(240.97+t_d)); % For t_d
e = Ew-A.*p.*(t_d-t_w);
% Partial water vapor pressures [hPa]
h = e./Ed.*100;
% Relative humidity
[%]
132
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Appendixes
% Reference saturated water vapor pressure [hPa]
Ed_r = (1.0007+3.46e-6*p_r)*6.1121*exp(17.502*t_r/(240.97+t_r));
e_r = Ed_r*h_r/100;
% Reference partial water vapor pressure [hPa]
T
= 273.15+t_d;
T_r = 273.15+t_r;
lam = lam/1000;
% Temperatures
[K]
% Reference temperature
[K]
% Instrument carrier wavelength [um]
% Group refractivity
N_g = 287.6155+4.88660/lam^2+0.06800/lam^4;
N_r1 = ((273.15/1013.25)*N_g*p_r/T_r)-11.27*e_r/T_r;
N_L = ((273.15/1013.25).*N_g.*p./T)-11.27.*e./T;
% For standard cond.
% For reference cond.
% For ambient cond.
% Group refractive index
n_r1 = N_r1/1e6+1;
% For reference conditions
n_L = N_L/1e6+1;
% For ambient conditions
% Computed vs. given reference refractive index (check)
n_diff = abs(n_r-n_r1);
% First velocity corrections
if n_r~=0
K_1 = d.*((n_r-n_L)./n_L);
else
K_1 = d.*((n_r1-n_L)./n_L);
end
d_1 = d+K_1;
d_1=d.*(n_r./n_L);
% From given n_r [m]
% From computed n_r1 [m]
% Corrected distances
[m]
% Direct computation of corr. distances (check)
%% 5) The vertical geometrical correction
h_s = h_s+hi;
h_t = h_t+hp;
% Station pillar height + instrument height [m]
% Target pillar height + prism height
[m]
for i = 1:length(d_1)
ds(i,1)=sqrt((d_1(i)^2-(h_t(i)-h_s(i))^2)/((1+h_s(i)/R)*(1+h_t(i)/R)))-d_1(i);
end
ds;
d_2 = d_1+ds;
% Vertical geometrical correction
% Corrected distances [m]
[m]
%% 6) The additive constant (Least-squares adjustment)
if ac0 ~= 0
d2 = d2+ac0/1e3;
end
[ac1,m_ac1] = ac(d_2);
d_3 = d_2+ac1;
[ac2,m_ac2] = ac(d_3);
% Applying initial additive constant
% Computation of additive constant [m]
% Control computation of additive constant [m]
if abs(ac2) > 1e-9
% Test if control additive constant is 0
error('Control computation of additive constant exceed the limit 1e-9.')
end
%% 7) The scale correction
if sc0 ~= 0
d3 = d3+sc0/1e6.*d3;
end
% Applying initial scale correction
133
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d_4 = d_3+(c_b+c_f)/1e3;
diff_d_4 = d_4-D;
% Applying projection corrections
% Differences in distances (measured-true) [m]
[sc1,m_sc1,res1] = sc(d_4,D);
d_5 = d_4+sc1.*d_4;
[sc2,m_sc2] = sc(d_5,D);
Appendixes
% Computation of scale correction [m/m]
% Control computation of scale correction [m/m]
if abs(sc2) > 1e-9
% Test if control scale correction is 0
error('Control computation of scale correction exceed the limit 1e-9.')
end
%% 8) The root-mean-square deviation (RMSD)
RMSD = sqrt((sum(res1.^2))/length(D));
% RMSD [m]
%% 9) The regression analysis
d_25 = d_2+(c_b+c_f)/1e3;
diff_d_25 = d_25-D;
% Applying projection corrections
% Differences in distances (measured-true) [m]
[alf,bet,m_alf,m_bet,RMSD_r,p,y,del,p3,y3,del3,RMSD_3] = reg(d_25,D);
RMSD_r;
% RMSD of simple linear regression
RMSD_3;
% RMSD of cubic regression
%% 10) Program outputs
% Write results to the source file sheet "results"
xlswrite(file,ac1*1e3,'results','B5:B5')
% Additive constant
[mm]
xlswrite(file,m_ac1*1e3,'results','C5:C5') % STD of additive constant [mm]
xlswrite(file,sc1*1e6,'results','D5:D5')
% Scale correction
[ppm]
xlswrite(file,m_sc1*1e6,'results','E5:E5') % STD of scale correction [ppm]
xlswrite(file,RMSD*1e3,'results','F5:F5')
% RMS deviation
[mm]
xlswrite(file,alf*1e3,'results','B6:B6')
xlswrite(file,m_alf*1e3,'results','C6:C6')
xlswrite(file,bet*1e6,'results','D6:D6')
xlswrite(file,m_bet*1e6,'results','E6:E6')
xlswrite(file,RMSD_r*1e3,'results','F6:F6')
%
%
%
%
%
Alfa coefficient
STD of alfa
Beta coefficient
STD of beta
RMS deviation
xlswrite(file,K_1.*1e3,'results','J5:J34')
xlswrite(file,ds.*1e3,'results','K5:K34')
xlswrite(file,diff_d_25.*1e3,'results','L5:L34')
xlswrite(file,-res1.*1e3,'results','M5:M34')
xlswrite(file,RMSD_3.*1e3,'results','F7:F7')
%
%
%
%
First velocity corr.
Ver. geometrical corr.
Distance differences
Residuals
% RMS deviation
% Create and save graphs to folder "Graphs"
graph(diff_d_25,diff_d_4,D,info,ac1,sc1,res1,p,y,del,p3,y3,del3)
% Display to the Matlab screen
T = datestr(now,31);
disp ' '
disp(['Date and time: ',T])
disp ' '
disp(['Program version: ',release])
disp ' '
disp 'Program terminated. All computations finished.'
disp ' '
disp 'Results have been written into the source file.'
disp 'Graphs have been saved into the working folder.'
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[mm]
[ppm]
[ppm]
[mm]
[mm]
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Appendixes
function [ac,m_ac]=ac(d)
% FUNCTION: Computation of additive constant
% INPUT:
d
- distances after velocity and geom. correction
% OUTPUT:
ac
- additive constant
%
m_ac - standard deviation of ac
% Vector of measurements L
L=d;
% Vector of unknowns X0
X0=[ L(1)
%0-24
L(7)
%24-72
L(13)
%72-216
L(19)
%216-432
L(25)
%432-864
0];
%c
% Vector L0
L0=[X0(1)
X0(1)+X0(2)
X0(1)+X0(2)+X0(3)
X0(1)+X0(2)+X0(3)+ X0(4)
X0(1)+X0(2)+X0(3)+ X0(4)+X0(5)
X0(1)
X0(2)
X0(2)+X0(3)
X0(2)+X0(3)+X0(4)
X0(2)+X0(3)+X0(4)+X0(5)
X0(1)+X0(2)
X0(2)
X0(3)
X0(3)+X0(4)
X0(3)+X0(4)+X0(5)
X0(1)+X0(2)+X0(3)
X0(2)+X0(3)
X0(3)
X0(4)
X0(4)+X0(5)
X0(1)+X0(2)+X0(3)+X0(4)
X0(2)+X0(3)+X0(4)
X0(3)+X0(4)
X0(4)
X0(5)
X0(1)+X0(2)+X0(3)+X0(4)+X0(5)
X0(2)+X0(3)+X0(4)+X0(5)
X0(3)+X0(4)+X0(5)
X0(4)+X0(5)
X0(5)];
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
0-24
0-72
0-216
0-432
0-864
24-0
24-72
24-216
24-432
24-864
72-0
72-24
72-216
72-432
72-864
216-0
216-24
216-72
216-432
216-864
432-0
432-24
432-72
432-216
432-864
864-0
864-24
864-72
864-216
864-432
% Vector l
l=L-L0;
% Plan
A=[1 0
1 1
1 1
1 1
1 1
1 0
0 1
0 1
0 1
0 1
1 1
0 1
0 0
matrix
0 0 0
0 0 0
1 0 0
1 1 0
1 1 1
0 0 0
0 0 0
1 0 0
1 1 0
1 1 1
0 0 0
0 0 0
1 0 0
A
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
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[m]
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0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
1
1
0
0
0
1
1
0
0
0
1
1
0
0
0
1
1
1
1
1
0
0
1
1
1
0
0
1
1
1
0
0
1
1
0
0
0
1
1
1
1
1
1
0
1
1
1
1
0
0
1
0
0
0
0
1
0
0
0
0
1
1
1
1
1
1
Aalto, 2012
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-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1];
% Weight matrix P
m0_apr=1;
% Apriori unit std
md=0.001/sqrt(5);
% Apriori std of measured distance
P=m0_apr^2.*diag(ones(1,length(L)).*md^2)^-1;
% Weight matrix
% Least-squares adjustment
N=A'*P*A;
n=A'*P*l;
x=N^(-1)*n;
X=X0+x;
ac=X(6,1);
% Additive constant
v1=A*x-l;
% 1st corrections
m0_apost=sqrt((v1'*P*v1)/(length(L)-5));
% Aposteriori unit std
if (m0_apost < m0_apr)
m0=m0_apost;
else
m0=m0_apr;
end
% Standard deviation of additive constant
Qxx=m0^2.*N^(-1);
Qx=sqrt(diag(Qxx));
m_ac=Qx(6,1);
function [sc,m_sc,res]=sc(d,D)
% FUNCTION: Computation of scale correction
% INPUT:
d
- distances after all corrections
%
D
- true distances
% OUTPUT:
sc
- scale correction
%
m_sc - standard deviation of sc
%
res - residuals
[m]
[m]
[m]
[m]
[m]
x = D;
y = D-d;
beta = sum(x.*y)/sum(x.*x);
% Manual computation
[beta,m_B,res] = regress(y,x,0.311); % Matlab function
sc=beta;
% Scale correction
m_sc=m_B(2)-sc;
% Standard deviation of scale correction
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function [alf,bet,m_alf,m_bet,RMSD_r,p,y,del,p3,y3,del3,RMSD_3]=reg(d,D)
% FUNCTION: Computation regression analysis
% INPUT:
d - distances after all corrections
[m]
%
D
- true distances
[m]
% OUTPUT:
a lot of staffs
L = d;
L0 = D;
l = L0-L;
n = length(L0);
%
%
%
%
Measured distances
True distances
Differences l
Number of distances
%% 1) Matrix computation (control computation)
A = [sum(L0.^2) sum(L0);sum(L0) 30];
b = [sum(l.*L0);sum(l)];
c = inv(A)*b;
% Coefficents
%% 2) Matlab fuctions (control computation)
[p,s]
= polyfit(L0, l, 1);
% Lin. regression
[y,del] = polyval(p,L0,s);
% Statistics
std = sqrt(diag(inv(s.R)*inv(s.R')).*s.normr.^2./s.df); % Std deviations
%% 3) By elements
D
= n*sum(L0.^2)-(sum(L0))^2;
% Denominator
alf = (sum(L0.^2) * sum(l) - sum(L0)*sum(L0.*l))/D; % Coefficient alfa [m]
bet = (n*sum(L0.*l)-sum(L0)*sum(l))/D;
% Coefficient beta [m/m]
for i = 1:n
v_y(i,1) = alf+bet.*L0(i)-l(i);
end
v_y;
% Residuals [m]
sum(v_y);
sum(L0.*v_y);
pom = sum(l.^2)-sum(l)*alf-sum(L0.*l)*bet;
sum(v_y.^2)-pom;
m_y
= sqrt((sum(v_y.^2))/(n-2));
RMSD_r = sqrt((sum(v_y.^2))/n);
Q11
Q22
m_alf
m_bet
=
=
=
=
sum(L0.^2)/D;
n/D;
m_y*sqrt(Q11);
m_y*sqrt(Q22);
%Check - should be 0
% Empirical deviation [m]
% Root-mean-square deviation (RMSD) [m]
% Standard deviation of alfa [m]
% Standard deviation of beta [m/m]
%% 4) Cubic regression
[p3,s3] = polyfit(L0, l, 3);
% Cubic regression
[y3,del3] = polyval(p3,L0,s3);
% Statistics
std3 = sqrt(diag(inv(s3.R)*inv(s3.R')).*s3.normr.^2./s3.df); % Std dev.
for i = 1:n
res3(i)=(p3(1).*L0(i)^3+p3(2).*L0(i)^2+p3(3).*L0(i)+p3(4))-l(i);
end
res3;
% Residuals [m]
RMSD_3 = sqrt((sum(res3.^2))/n);
% RMSD [m]
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function graph(diff_d_25,diff_d_4,D,info,ac,sc,res,p,y,del,p3,y3,del3)
mkdir 'Graphs'
cd './Graphs'
%% The additive constant
figure(1)
%set(1,'Visible','off')
hold on, grid on, box on
title({['\bfAdditive constant analysis\rm'];[char(info(1)),' +
',char(info(2))];[char(info(3)),', ',char(info(4)),', ',char(info(5))]})
xlabel('\bfDistance [m]')
ylabel('\bfDifference (measured-true) [mm]')
scatter(D(1:5),diff_d_25(1:5).*1e3,'ob','filled')
scatter(D(6:10),diff_d_25(6:10).*1e3,'og','filled')
scatter(D(11:15),diff_d_25(11:15).*1e3,'or','filled')
scatter(D(16:20),diff_d_25(16:20).*1e3,'oc','filled')
scatter(D(21:25),diff_d_25(21:25).*1e3,'om','filled')
scatter(D(26:30),diff_d_25(26:30).*1e3,'ok','filled')
fplot(@(x) -ac*1e3,[0,900],'b')
if sc>1e-6
axis([0,900,-2.5,+0.5]) %Axis for Leica TCA2003
else
axis([0,900,-1.0,+1.0]) %Axis for Kern Mekometer ME5000
end
set(gca,'xtick',[0 72 216 432 648 864])
legend('From 0','From 24','From 72','From 216','From 432', 'From
864','Location','NorthEast')
print('-dpng','-r300',[char(info(4)),'_1_ac']);
%% The scale correction
figure(2)
title({['\bfScale correction analysis\rm'];[char(info(1)),' +
fplot(@(x) -sc*1e3*x,[0,900],'b')
if sc>1e-6
else
end
set(gca,'xtick',[0 72 216 432 648 864])
legend('From 0','From 24','From 72','From 216','From 432','From
print('-dpng','-r300',[char(info(4)),'_2_sc']);
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%% Residuals
figure(3)
title({['\bfResiduals analysis\rm'];[char(info(1)),' +
ylabel('\bfResidual (corrected-true) [mm]')
scatter(D(1:5),res(1:5).*1e3,'ob','filled')
scatter(D(6:10),res(6:10).*1e3,'og','filled')
scatter(D(11:15),res(11:15).*1e3,'or','filled')
scatter(D(16:20),res(16:20).*1e3,'oc','filled')
scatter(D(21:25),res(21:25).*1e3,'om','filled')
scatter(D(26:30),res(26:30).*1e3,'ok','filled')
if sc>1e-6
else
end
set(gca,'xtick',[0 72 216 432 648 864])
legend('From 0','From 24','From 72','From 216','From 432','From
print('-dpng','-r300',[char(info(4)),'_3_res']);
%% Simple linear regression
figure(4)
title({['\bfSimple linear regression\rm'];[char(info(1)),' +
plot(D,-(y+2.*del).*1e3,'r+')
plot(D,-(y-2.*del).*1e3,'r+')
refcurve(-p.*1e3)
if sc>1e-6
else
end
set(gca,'xtick',[0 72 216 432 648 864])
legend('From 0','From 24','From 72','From 216','From 432','From 864','P=95% interval','Location','NorthEast')
print('-dpng','-r300',[char(info(4)),'_4_sim']);
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%% Cubic regression
figure(5)
title({['\bfCubic regression\rm'];[char(info(1)),' +
plot(D,-(y3+2.*del3).*1e3,'r+')
plot(D,-(y3-2.*del3).*1e3,'r+')
refcurve(-p3.*1e3)
if sc>1e-6
else
end
set(gca,'xtick',[0 72 216 432 648 864])
legend('From 0','From 24','From 72','From 216','From 432','From 864','P=95% interval','Location','NorthEast')
print('-dpng','-r300',[char(info(4)),'_5_pol']);
cd '../'
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Appendix 8: Graphs – 2011-10-06, Leica TCA2003
141
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Appendix 9: Graphs – 2011-10-07, Leica TCA2003
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Appendix 10: Graphs – 2011-11-24 and 25, Kern Mekometer ME5000
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Appendix 11: Graphs – 2011-11-25 and 28, Kern Mekometer ME5000
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Appendix 12: Example calibration certificate issued by the FGI
GEODEETTINEN LAITOS
KANSALLINEN MITTANORMAALILABORATORIO
GEODETISKA INSTITUTET
NATIONELLT MÄTNORMALLABORATORIUM
FINNISH GEODETIC INSTITUTE
NATIONAL STANDARDS LABORATORY
PL 15 (Geodeetinrinne 2), FI-02431 MASALA, FINLAND
KALIBROINTITODISTUS
KALIBRERINGSBEVIS
Certificate of Calibration
Numero - Nummer:
Number
20 / 2009
Sivu - Sida
Page
Tilaaja - Uppdragsgivare:
Customer
Zhengzhou Surveying and Mapping Institute.
Kalibroitu laite - Kalibrerat instrument:
Calibrated instrument
Electronic distance meter.
Valmistaja - Tillverkare:
Manufactured by
Wild.
Tyyppi - Typ:
Model
DI2002.
Sarjanumero - Serienummer:
Serial number
180495.
Kalibrointipäivä - Kalibreringsdatum:
Date of calibration
June 23–25, 2009.
Päiväys - Datum:
Date
July 3, 2009.
1(2)
Allekirjoitukset - Underskrifter:
Signatures
Director General
Risto Kuittinen
Specialist reseach scientist
Jorma Jokela
Liitteitä - Bilagor:
Documents attached
–
A National Standards Laboratory is responsible for the maintaining of a national standard and its traceability to SI-units. The
Laboratory is also responsible for the dissemination of the units to other reference standards. As prescribed in the Law no.
581/2000, the Finnish Geodetic Institute maintains standards for measurements in geodesy and photogrammetry, and acts as
the National Standards Laboratory of Length and Acceleration of Free Fall.
The Certificate may not be reproduced other than in full, except with the prior written approval of the Finnish Geodetic Institute.
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GEODEETTINEN LAITOS
Finnish Geodetic Institute
PL 15 (Geodeetinrinne 2)
FI-02431 MASALA
FGI Certification of Calibration no. 20 / 2009, page 2(2)
Calibration of electronic distance meter Wild DI2002 n:o 180495 at
the Nummela
Standard Baseline of the Finnish Geodetic Institute on June
23–25, 2009.
Measured
distances:
Three calibrations in three days, in each of them all the 15 pillar intervals
24 m to 864 m have been measured from both ends. EDM instrument and
prism reflectors fixed on observation pillars using permanently installed Kern
forced-centring plates and adapter plates with 5/8” fixing screws. Measurements with two precise prism reflectors. Measurements were performed by
Messrs. Xue Ying, Fu Ziao and Fan Gang, assisted and guided by Mr. Jorma Jokela.
Other measurements:
Temperature observations with psychrometers at the EDM instrument (dry,
wet) and at the reflector (dry). Air pressure measurements with an aneroid
barometer at the EDM instrument. W eather was during calibrations clear
and sunny, temperature +18.7 °C – +23.6 °C, air pressure 101.32 kPa –
101.60 kPa, relative humidity 24 % – 62 %.
True distances:
Interference measurements of the Nummela Standard Baseline in autumn
2007. Projection measurements at the baseline between underground
benchmarks and observation pillars on May 11 – 14, 2009, by Mr. Joel Ahola
and Ms. Sonja Nyberg, and on June 29 – July 1, 2009, by Mr. Jorma Jokela
and Ms. Sonja Nyberg.
Computation:
Weather corrections, vertical geometrical corrections, projection corrections.
Additive constant is determined as an unknown in the first adjustment. Scale
correction is determined in a least-squares adjustment by fitting the corrected
observed values with the true distances. Results (additive constant and scale
correction, 1-σ uncertainty) of the three calibrations were:
June 23: –0.478 mm ±0.131 mm ja +0.424 mm/km ±0.146 mm/km,
June 24: –0.430 mm ±0.085 mm ja +0.802 mm/km ±0.100 mm/km.
June 25: –0.211 mm ±0.062 mm ja +0.245 mm/km ±0.078 mm/km.
The result of calibration is unweighted average of these. Based on interference measurements and projection measurements, standard uncertainty of
true values is estimated to ±0.15 mm/km, which is included in the total uncertainty of result of calibration. Previous experience is also used in estimation of total uncertainty.
Result of calibration, with
extended (2-σ)
uncertainties:
Additive constant: –0.37 mm ±0.16 mm.
Scale correction: +0.49 mm/km ±0.44 mm/km.
Used methods meet the requirements of ISO 9001 and ISO 17025 quality
standards
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147

MASTER`S THESIS

Transcription

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